MEASURINGPROXIMITYINDUCEDEFFECTSINTOPOLOGICALINSULATORSUSINGSCANNINGPROBEMICROSCOPYTECHNIQUESByIanMatthewDaytonADISSERTATIONSubmittedtoMichiganStateUniversityinpartialentoftherequirementsforthedegreeofPhysics|DoctorofPhilosophy2016ABSTRACTMEASURINGPROXIMITYINDUCEDEFFECTSINTOPOLOGICALINSULATORSUSINGSCANNINGPROBEMICROSCOPYTECHNIQUESByIanMatthewDaytonScanningprobemethodssuchasscanningtunnelingmicroscopy(STM)andsubsurfacechargeaccumulationimaging(SCAI)areverypowerfulandimportanttoolstoinvestigatetheelectronicneartheinterfacesofexoticmaterials.Forthisdissertation,theprimarymaterialstudiedisathree-dimensionaltopologicalinsulator,whichhasbeendiscoveredinrecentyearstoharboragreatdealofnovelphysicalphenomenon.Theprimaryfeatureofnotewiththesematerialsistheirintrinsicspin-orbitcoupling,whichgivesrisetothesurfacestateswherethespinandmomentumoftheconductingelectronsarelockedinsurprisingwaystooneanother.Sp,thedispersiondiagramshowstwobrancheswithlinearenergyversuswavenumbercrossingthesemiconductorgaptoformaDiracconefeature;theseconductingstatesresideonthesurfaceandareknownastopologicalsurfacestates(TSS).Thisstateistopologicallyprotectedbytime-reversalsymmetryinthattheseconductingsurfacestatesarerobustagainstscatteringfromnon-magneticdefects.Inthisdissertation,IwillpresentSTMresultsforsuperconductor/topologicalinsulatorinterfaces,inwhichthesuperconductorinducessuperconductivityintothetopologicalsurfacestates.SuperconductivityiscausedbythepairingofelectronsintobosonswhichareknownasCooperpairs,andisyetanotherphenomenonthatinvolvesratherintricatespinandmomentuminteractions.Inthesemeasurements,weobservedoscillatorybehaviorneartheinterfacesaswellasasurprisingreverseproximityinwhichthetopologicalsurfacestatesappeartoleakintothesuperconductingmaterialaswell.IntheAppendix,IwillalsopresentSTMandSCAIresultstakenatconventionalinsulator/topologicalinsulatorinterfaces,wheretheTSSispredictedtochangelocationsbasedontheconventionalinsulatormaterialdepositedonthesurfaceinanknownasthe\dual"proximityInparticular,weinvestigatetheinterfaceofZnSeandBi2Te3.Whiletherehasbeennotheoreticalpredictionforthisspinterface,ourresultsshowthattheTSSremainsattheinterfaceanddoesnotchangelocationasafunctionofbiasvoltage.Eventhoughtheseresultsarepreliminary,theysetthestageforinterestingandimportantmeasurementsinthefuture.ThisdissertationisdedicatedtomywifeChristinaDaytonandtomyparentsTimothyandDarylDayton.Withouttheirendlessloveandsupport,noneofthiswouldhavebeenpossible.ivACKNOWLEDGMENTSIwouldliketothankmyadvisorDr.StuartTessmerforhisexceptionalmentoringandteachingduringmytimeingraduateschool.Yourkindandsteadyguidancewasandiscrucialtomysuccess,andIhavelearnedmorethanIcouldeverhaveimaginedunderyourtutelage.Itrulycouldnotimagineamoreperfectforanadvisorandteacher,andIamveryproudtohavebeengiventheopportunitytoworkforyou.NextIwouldliketothankDr.RezaLoloeeforallofhishardworkandhelpthathehasgivenmethroughouttheyears.Thereisalwaysagreatdealtolearninthisespeciallyintermsoftexperimentalskillsets,andyoutaughtmeanincredibleamount.Ofcourse,IalsowouldliketothankDr.NormanBirge,Dr.AlexLevchenko,Dr.ScottPratt,andDr.JoeyHustonfortheirimmeasurablehelpandsupportwhileservingonmygraduatecommitteeduringmytimeatMSU.Dr.AlexLevchenkoandDr.NicholasSedlmayrplayedaverytroleinthetheoreticaldiscussionsandcalculationsthatprovidedinterpretationanddetailedmodelingofmydata.Iowethemagreatdealofgratitudefortheirhardworkandhelpfuldiscussionsonthematerialpresentedinthisdissertation.Duringmytimeingraduateschool,Imademanyfriendswhohelpedmealongtheway,notonlywithtechnicaldiscussions,butalsobybringingnosmallmeasureoflevitytoworkinginBPS.Dr.EricGingrich,Dr.BillMartinez,Dr.SeanWagner,BethanyNiedzielski,VictorAguilar,MatthiasMuenks,ConnorGlosser,SamLipschutz,MattMondragon,JaredDoster,andChrisWillis;thankyousomuchforyourfriendshipandhelpthroughouttheyears.Youmadeworkinghereamuchmoreenjoyableexperience.ThankyoutoEricGoodwin,VictorRamirez,andMichaelGottschalkforyourcompan-vionshipandhelpinthelab.IhopethatIwasabletoteachyouallsomethingalongthewayandIlookforwardtohearingaboutthegreatthingsthatyouwillaccomplishinyourlives.Iwouldliketothankmyparents,TimothyandDarylDayton,foralloftheirloveandsupport.Attheendoftheday,everythingIhaveandwillaccomplishisbecauseofyou,andItrulyhaveanimmeasurableamountofgratitudeforeverythingyouhavedoneforme.Lastbutnotleast,IwouldliketothankmylovingwifeChristinaDayton.Youseeminglyappearedoutofnowhereinthemiddleofmygraduatestudiesandhavemademyalreadybrightworldmorebrilliantthanithaseverbeen.Wetakethisnextsteptogether,andIcannotwaittoseewhatlifehasinstoreforus.Thankyou,oneandall,formakingthispossible.TheworkinthisdissertationwascompletedwithsupportfromtheDepartmentofEd-ucationthroughGAANNAwardNo.USDEP200A140215andfromthePeterSchroederEndowment.viTABLEOFCONTENTSLISTOFTABLES....................................ixLISTOFFIGURES...................................xChapter1Introduction...............................1Chapter2ExperimentalTechniques.......................42.1ScanningTunnelingMicroscopy.........................42.1.1TunnelingTheory.............................42.1.2ScanningTunnelingSpectroscopy....................62.1.3ExperimentalSetup............................92.1.3.1BesockeDesignSTM......................92.1.3.2CryogenicSTMSystem....................102.2ScanningChargeAccumulationImaging....................122.3TipPreparation..................................19Chapter3TheTheoriesofTopologicalInsulatorsandSuperconductivity223.1TopologicalInsulators..............................223.1.1QuantumHallandQuantumSpinHall..........243.1.23DTopologicalInsulatorsandBi2Se3..................273.2Superconductivity.................................313.2.1BCSTheory................................323.2.2P-WaveSuperconductors.........................373.2.3SuperconductingProximityct....................39Chapter4ProbingSuperconductor/TIInterfaces...............434.1SuperconductingProximityofPbBionBi2Se3.............434.1.1Motivation.................................434.1.2Introduction................................464.1.3Experiment................................494.1.4TheoryandCalculations.........................584.1.5Conclusions................................604.2SuperconductingProximityofPureNbonBi2Se3...........624.2.1Introduction................................624.2.2Results...................................634.2.3Conclusion.................................74Chapter5Conclusions................................775.1SummaryofResults...............................775.2FutureGoals...................................77viiAPPENDICES......................................80AppendixASuperconductingProximityofPbonBi2Se3..........81AppendixBTopologicalInsulator/ConventionalInsulator\Dual"Proximity85AppendixCKelvinProbe...............................98BIBLIOGRAPHY....................................102viiiLISTOFTABLESTable4.0a:TablepresentingthecalculatedvaluesofandotherparametersforthebestDOScurvesatvariousdistancesfromthesuperconductingisland.Thealgorithmutilizedboththermalandenergeticsmearing.HereisthecharacteristicenergysmearingofthematerialandListherelatedmeanfreepath,andthetwovaluesarerelatedbytheequationL=hvf=Allcalculationsweredoneusingatemperatureof4.2K.Thep-wavecalculationsweredoneviaasumofseparates-wavesuperconductingcurvesweighedaccordinglybyap-waveorbitalfunction.55Table4.0b:AcomparisonofthegapwidthstakenfromthebestDOScurveswiththewidthsextractedusingourlinear-subtractionmethod.........55TableB.1:TablesummarizingtheresultsoftheWuet.al.paperandlabelingthecharacteristicpropertiesofthematerialsusedinthecalculations.Vgisthebandgapwidth,istheworkfunction,aisthebulklatticeconstant,aisthelatticemismatch,EbisthebindingenergybetweentheCIandTIlayers,andZisthelocationoftheTSSforthespgeometrycalculated.[2].................................88ixLISTOFFIGURESFigure2.1:Vacuumbarrierbetweenatipandsample,withtheelectronwavefunctionshowninred.Thisdemonstratesthatapropagatingelectronhastheabilitytotunnelfromthesampletothetipwhenavoltageisappliedbetweenthem.Here,eVisthecontributionfromthebiasvoltage....5Figure2.2:Bysweepingthebiasvoltage,weareableto(empty)theempty(occupied)statesofthesample.Asthetunnelingcurrentdependsonthenumberofaccessiblestates,thisallowsustoextractitsLDOS......7Figure2.3:(a)SchematicshowingthelayoutofaBesockedesignSTM.(b)Actualhouse-builtSTMintheTessmerlab.....................9Figure2.4:ThefullmeasurementsetupwithSTMdipperfullysubmergedintheblueliquidHeliumdewar...........................11Figure2.5:Diagramofcryostatusedinourlab,showingalloftheimportantchambersandfeatures.Here,thesorptionpump(Sorb)isintheshapeofatorusandiffoundinsideofthevacuumchamber,whichallowstheSTMtopassthroughitandreachthebottom....................13Figure2.6:AhighlydiagramdemonstratingtherelationshipofgeometriccapacitanceCGonthedistancebetweenthetipandthesample.....15Figure2.7:(a)CircuitdiagramofourSCAImeasurementchip.TheHEMTactsasahighlysensitivechargedetectorthatisdirectlyconnectedtothescanningprobetip,whichallowsustoseechargeuctuationsinthesampleduetoimagechargesformedinthetip.(b)diagramdemonstratingtheexistenceofavoltagedividerfortheACexcitationvoltagegoingthroughtwocapacitors............................16Figure2.8:TheleftimageshowsafullyassembledSCAImeasurementchipcompletewithprotectionloops.ThemethodofassemblyutilizestweezerworkwithconductiveepoxytofastentheprimarycircuitelementsandconnectionwirestotheGaAschip.Inordertoconnectthegoldpadstothecorrectelements,weutilizeagoldwirebonder.TherightimageshowsaHEMTwithnoconnections.TheHEMTasawholeisapproximately600mwideandthegatepaditselfisonly80mwide..........18xFigure2.9:SEMimagesdemonstratingthetipgeometryaftertheroughetchingstep.Thebulbattheendwillideallydropduetogravityassistanceduringtheetchingprocess.(a)Initialscanwherethebulbisclearlyvisibleand(b)Closerlookatthesametip.....................20Figure2.10:SEMimagesdemonstratingwhathappenstoanSTMtipwhenitisdropped.Whatwasonceanidealcandidateforuseasascanningprobeisnowrenderedunusableduetothelargeradiusofcurvatureatthebendinthewire.(a)Initialscanwherethebendisclearlyvisible.(b)Closerlookatthesametip.............................20Figure2.11:SEMimagesshowingtheofasinglecuttip.(a)Whilethecuttiplookstohavemultiplepoints,thereisoneclearpointthatismorethan50furtheroutfromtheothers.(b)Closeupoftheapexshowingthattheradiusofcurvatureislessthan40nm...............21Figure3.1:AdiagramofthequantumHallsystemwithintegerLandaulevelsintheinteriorintheclassicalpicture(a)andtheresultingquantumbehavior(b).Theedgeofthetwo-dimensionalsystemconductswhilethecenterremainsinsulatingduetothelocalizedstatesoftheelectrons.Figureadaptedfrom[6]...........................25Figure3.2:Thisdiagramdemonstratesthetwospinpolarizedconductingstatesontheedgeofthetwo-dimensionalsystem,wheretheredlinecanbeconsideredtobespin-upelectronsandthebluecanbeconsideredtobespin-downelectrons.Figureadaptedfrom[6]................26Figure3.3:TheDiracconeisaconsequenceofthespin-orbitcouplingfrominTI's.Theredandbluecrossingchannelscorrespondtothespin-upandspin-downchannelsrespectively.ThecrossingpointinthecenterisknownastheDiracpoint,andistypicallynotfoundattheFermilevelEFofthematerial.Figureadaptedfrom[6]......................27Figure3.4:(a)SchematicofthestructureforBi2Se3.Bi2Se3isalayeredmaterialconsistingofstackedquintuplelayersmadefrom3Seatomsand2Biatoms.ThelayersareattractedtooneanotherthroughvanderWaalsforcesandarethereforeasytoseparatebyexfoliation.(b)Schematicshowingthep-orbitalsplittingofBiandSeneartheFermilevel,whichisdenotedbythedottedblueline.Region(I)takesthechemicalbondingofBiandSeintoaccount,whileregion(II)furtherdemonstratestheontheenergylevelsfromthecrystalld.Finally,region(III)showstheofstrongspinorbitcoupling,inwhichbandinversionoccursandthetopologicalphaseiscreated.(a)isadaptedfromRomanowichet.al.and(b)isadaptedfromQiandZhang.[18,17]..............30xiFigure3.5:(a)DiagramoftheDiracconewithspinpolarizedconicalsectionsink-space.AtenergiesabovethecrossingpointoftheDiraccone,theelectronswillbepolarizedinonedirectionwhentravelingaroundthecone,whereasatenergiesbelowthepoint,electronswillbepolarizedintheoppositedirection.(b)BasiccalculationdemonstratingtheexpectedbehavioroftheDiracconewhenlookingatthedensityofstates.TocalculatetheexpectedDOScurve,weneedtointegrateovertheFermisurfaceintwo-dimensions.Theresultingintegralgivesalineardispersionrelationink,whichyieldsalinearDOSversusenergycurve.ThisiswhatwetheoreticallyexpecttoseewhentakingscanningtunnelingspectroscopyoverabareTI.................................31Figure3.6:Astheelectronmovestotheright,itcausesadistortioninthelattice,creatingaregionwithhigherpositivecharge.Thisregionattractsasecondelectronintothenewlyformedpotentialwell,andthetwoelectronsbecomeapairknownasaCooperpair..............34Figure3.7:Acalculatedsuperconductingenergygapdemonstratingwhattheex-pecteddensityofstateswouldlooklikeonaBCSsuperconductor.Thewidthofthegapis0.Asmallamountofthermalbroadeningwasappliedinthecalculationtopreventthecoherencepeaksfromgoingtoy.....................................36Figure3.8:(a)Aplotofthesuperconductingpairingpotentialk)forap-wavepairingsymmetryinthekxkyplane.(b)Thedensityofstatesofapurep-wavesuperconductor.TheDOSisnotfullygappedlikeinthes-wavecase,whichresultsina\v"shapeinsideofthesuperconductinggap.39Figure3.9:Animagedemonstratingthebehaviorofr)asafunctionofthepaircorrelationamplitudeF(r)andtheinteractionparameterg(r)......42Figure4.1:Proposedanti-dotgeometrywhereacontinuoussuperconductingisdepositedonthesurfaceofaTIwithanti-dotsspacedevenlythroughoutthewhichwouldgoallthewaytotheTIsurface.ByapplyingamagneticnormaltotheS/TIsurface,Majoranaboundsstatesaretheoreticallypredictedtoformaroundtheboundaryoftheanti-dot,whichcouldbeprobedusingSTS.FigureadaptedfromtheHasanandKanecolloquiumontopologicalinsulators.[16]..............44Figure4.2:LDOStaken40nmfromasuperconductingislandat4.2Kwiths-waveandp-waveDOScalculatedcurves.Atthistemperature,itistoseemuchbetweenthetwocalculatedcurvesasbroadeningcausethemtoappearverysimilar.FittingparametersareshownlaterinTable4.0a..............................45xiiFigure4.3:(a)Aschematicofthestandardproximitypictureshowingtheinducedsuperconductingenergygap[61],ataninterfaceasafunctionofpositionx,whereFisthesuperconductingcondensateamplitude.Thepairinginteractionconstant,g,isgenerallytakentohavetheformofaclearstepfunction,butatsmallscales,thestepactuallyhasaslopeduetoelectronicscreening[62].(b)AschematicillustrationforthegeometryoftheSTMprobescanningoverthesurfaceofTIBi2Se3inproximitytoaPbBiisland.Inthisexperiment,weareinterestedinmeasuringthesuperconductingcoherencelengthintheplaneparalleltotheTIsurface,notinthenormaldirectiontothesurfaceasdoneinmanyotherexperiments...............................48Figure4.4:(a)STStakenona250nmthickofPb0:3Bi0:7depositedonBi2Se3andwithacalculateds-waveDOSthathasbeenappropriatelybroadenedtoaccountforthermalandscattering(b)Four-probemeasurementtakenona400nmthickofPb0:3Bi0:7takenbyDr.RezaLoloee.Thetransitiontemperatureof8.3Kcloselymatchesthenominalvalueof8.2K.[64]Bothmeasurementstakenat4.2K...............50Figure4.5:EDSmeasurementstakenonour250nmthickPbBishowingtheactualatomiccompositionofouralloy.EDSshowedthattheactualcompositionwas36%Pband64%Bi,whichiswellwithintheexpectedcompositionrangeforoursuperconductingislands.............51Figure4.6:Imagetakenunderopticalmicroscopeshowingthesuperconductingarraypatternafterthermalevaporation.ThissamplewasexposedtoairandwasthereforenotmeasuredunderSTM...................52Figure4.7:(a)and(b)Atomicforcemicroscopy(AFM)topographsofathermallydepositedPbBidot.Wecanclearlyseeaverygrainyappearancetotheoveralldot,butuponcloserinspectionitisclearthatthedotiscomprisedofmanysmallsuperconductingdroplets.(c)STMtopographofaPbBidropletwithitsrespectiveheighttraceshownin(d).TheradiusofcurvatureatthebaseofthedropletisanartifactduetotheradiusoftheSTMtip.TheactualinterfacebetweenthedropletandtheTIisverysharp,whichisidealforourmeasurement.Scalebars:(a)500nm(b)100nm(c)40nm.............................53xiiiFigure4.8:TheleftpanelrepresentsdI=dVcurvesmeasuredat4.2KtakenatvariousdistancesfromaPbBiisland.TheLDOSdisplaysclearsignatureoftheinducedsuperconductinggap,withthearrowsdenotingthelocationofthecoherencepeaks.TheonlyfeatureatdistancesfarfromthesuperconductingislandistheDiraccone.Anothernotablefeatureofthepresenteddataisvisibleoscillationsoutsideofthesuperconductinggap.TherightpanelrepresentsdI=dVcurvesmeasuredontislandsatnominallythesameconditions.ThedatashownheredemonstratesanincreasingDOSforbothpositiveandnegativevoltagesoutsideofthegap,asopposedtotheatDOSpredictedbyBCS.ThisisindicativeofDiracconestatesinthesuperconductor.AlldataarenormalizedsothatdI=dVj20mV=1..............................56Figure4.9:TotalenergygapwidthmeasuredbySTMat4.2KwiththeexpectedexponentialdecaydescribedbythefunctionGapWidth=20ex=˘,demonstratingthatastheprobemovesfurtherfromthePbBidroplet,thesuperconductinggapdecreasesinwidth.Herethebestgivesapairingpotentialof0=3:640:40meVandacoherencelengthof˘=540200nm.Inordertoextractforeachdatapoint,wetooklinearfrom-6{-1mVand1{6mVofthedI=dVcurveandsubtractedthemfromthefulldatainordertoenhancethecoherencepeaksonbothsidesoftheFermilevel.Theinsetshowsthedataandcurveoveranexpandeddistancerange...............57Figure4.10:(a,b)Thelocaldensityofstatesintheelimitasafunctionofenergy(a)attdistancesfromtheboundaryof0:5R;1R;1:5R,shownbythered,black,andbluecurvesrespectively,andasafunctionofposition(b)atenergies0:1ETh;0:8ETh;1:5ETh,onceagainshownbythered,black,andbluecurvesrespectively,withlinesrepresentingthelinearizedanalyticalresult,andsymbolsthenumericalresultoffullnonlinearEq.(4.1).Theinsetshowacomparisonbetweenthegapfoundexperimentally(bluediamonds)andtheanalyticalresult(solidline).(c)ThelocaldensityofstatesasafunctionofenergyforthetopologicalinsulatorsurfacestatestotheleftoftheTI/Sboundaryat0,whichistreatedasbulksuperconductor.(d)ThelocaldensityofstatesasafunctionofenergyforthetopologicalinsulatorsurfacestatestotherightoftheTI/Sboundaryat0,whichistreatedasbaretopologicalinsulator.Aphenomenologicaldampinghasbeenincluded,andthebulkdensityofstatesareincludedasacomparison.....................61xivFigure4.11:(a)AschematicillustrationforthegeometryoftheSTMprobescan-ningoverthesurfaceofTIBi2Se3inproximitytoaNbisland.(b)Aschematicofthestandardproximitypictureshowingtheinducedsuperconductingenergygap[61],ataninterfaceasafunctionofpo-sitionx,whereFisthesuperconductingcondensateamplitude.Thepairinginteractionconstant,g,isgenerallytakentohavetheformofaclearstepfunction,butatsmallscales,thestepactuallyhasaslopeduetoelectronicscreening[62].......................63Figure4.12:SEMimagetakenbycollaboratorCanZhangshowingaregionofNbarrays.Inthissampleweareprimarilygivensquareandhexagonalarrayswithdistanceinbetweenislands.Thesamplealsocontainsanti-squarearrays,circulararrays(likethoseusedpreviouslyinourPbBiandPbmeasurements),tri-junctions,andloopsjunctions.TheloopjunctionswouldallowustothreadamagneticthroughasuperconductingloopandmeasuretheresultfromtheinducedphaseintheNbjunction.ThearraysarenecessaryinordertoeliminatetheneedleinthehaystackprobleminherentinSTMmeasurements.............64Figure4.13:AFMtopographtakenbycollaboratorCanZhangshowinganarrayofhexagonalNbislands.ThisdemonstratessmoothNbislandswithacleanTIsubstrate..................................65Figure4.14:MeasurementoftheDiracconetakenfarfromtheNbarrays.Thisspectrumwastakenat1.78K.Webelievethattheconeisnotperfectlysharpduetothesmallamountofthermalbroadeningatthistemperature.66Figure4.15:Proximitymeasurementtakennear1mofaNbarray;thedistancesfromtheedgeofaNbislandareindicatedintheplot,wheretheredarrowsnotethelocationofthecoherencepeaks.Wecanclearlyseethegapdecreaseinwidthaswemoveawayfromtheisland,thenbegintoincreaseagainaswearemostlikelygettingclosertoanotherarray.Measurementtakenat1.6K.........................68Figure4.16:TwospectratakenattpointsdemonstratingTomasch/Friedel-likeoscillationsoutsideofthesuperconductinggap.Interestingly,thetopcurvelooksliketheoscillationsaresimplysuperposedonanormalDiraccone,butsincesuperconductivitymustbepresentinorderforthetobeobserved,itispossiblethatthismeasurementwastakennearaweakNbisland.Measurementsweretakenat1.6K............69xvFigure4.17:VariousspectratakenonerentNbislandsdemonstratingtheinverseproximityoftheTSSonthesuperconductor.Intwocurves,weappliedusingcalculateds-waveDOScurvesthatbeenthermallybroadenedandhadscatteringapplied,wherethevaluesoftheseparametersareshownintheplot.Thecalculatedcurveshadatemperatureof1.6K,acoherencelengthof38nm,andathicknessof60nm.Surprisingly,wecannotseeclearcoherencepeaksinthesuperconductinggapspotentiallyduetotheinducedTSSformingp-wavesuperconductivityintheNbislands.Allmeasurementsweretakenattemperaturesbetween1.5-1.8K,wherethesuperconductinggapisexpectedtobebothverydeepandtohavesharpcoherencepeaks.ThetopcurveisaDiracconemeasuredinaregionofthesamplewherenoNbislandshadbeendeposited........................70Figure4.18:Room-temperaturespectratakenonandNbislands.The3curvesontheleftweretakenonNbislandsfoundinthesamearraywhilethecurvesontherightweretakenat3separatelocationsonbareBi2Se3.WebelievethatthisisthesignatureoftheDiracconeonNbatroomtemperature,whichindicatesthatthisinverseproximityislikelynotdependentonthesuperconductingstateoftheNbandthereforemuststemfromsomeothermechanism......................71Figure4.19:LDOSdatawiththeoreticalcurvesbasedonamodelthatincludesnativemetallicstatesandproximitycoupledTIsurfacestates.Model1(blue)hasnoBCSpairing.Model2(purple)hasBCSpairinginthenativemetallicstatesonly.Model3(red)haspairinginboththenativeandtheproximity-coupledTIstates.......................74Figure4.20:ExperimentaldataoftheDOStakenon3separateislandswithcalculateddemonstratingtheleakageoftheTSSintothesuperconductingNbislands.(a-c)allutilizevF,,andabroadeningtermasfreeparameterswhile(d-f)allowtobeafreeparameterinthecalculationaswell.Ingeneral,(d-f)havebetterasitisexpectedthatwillbebyboththeproximitytotheTImaterialaswellastheTSSstatesleakingintothesuperconductor...........................75Figure5.1:SchematicsoftheproposedmeasurementwhichwillutilizeSCAImeasure-mentstomeasurethepathoftheJosephsoncurrentinabiasedS/TI/Sjunction....................................79FigureA.1:STStakenona400nmthicklayerofPbdepositedonthesurfaceofBi2Se3.Herewecanseeaveryclearsuperconductinggapaswellassignaturesofphononoscillationmodesoutsideofthesuperconductinggap.Thismeasurementwastakenat4.2K.................82xviFigureA.2:AFMtopographofPbislandsdepositedonthesurfaceofBi2Se3.Theislandsaregrainy,muchlikethePbBisamples,butalsoareheavilyoxidizedduetotheintrinsicnatureofPbexposedtoair.........83FigureA.3:STMdatatakenonasamplemadewithPb/Bi2Se3.Whilewecanseeaprettycleaninterface,theSTSdatashowinducedgapsthatarefartoowidetobesuperconducting.WebelievethatthesegapsstemfromchargedensitywavescausedbylatticemisalignmentoftheTIcrystal.Weexcludedatafromposition4intheimageduetoveryhighmechanicalinstabilityattheinterface..........................84FigureB.1:FiguretakenfromWuet.al.demonstratingthepossiblebehavioroftheTSSwhenaCIinputincontactwithaTI.TheTIwilleither\topologize"theCI,causingtheTSStotothesurface,becometrivializedbytheCI,pushingtheTSSdownoneQL,orthematerialswillnotoneanotherinanymeaningfulway,leavingtheTSSattheinterface.[2]...86FigureB.2:DFTcalculationtakenfromWuet.al.showingthebandstructuresofdif-feringTI/CIinterfaces,spZnS/Bi2Se3(a-d)andZnSe/Bi2Se3(e-h).Thedotsindicatethespectralweightsandcontributionsfromatoms,denotedbythesizeandcolorrespectively.DPUandDPLdenotetheDiracpointsoftheupperandlowerssurfacesrespectively.[2]....87FigureB.3:(a)AFMphaseretracedemonstratingtmaterialsonthesurfaceofabakedsample.(b)STMtopographofthesample,demonstratingthesamefeaturesseenintheAFMscanshownin(a).............90FigureB.4:Diagramshowingsampleafterbakingandassembly.Thesampleitselfisusuallyaround3mminsizeinordertoallowforalargeareatoscanusingoursurfaceprobes...........................90FigureB.5:(a)TheelectricbetweenthetipandtheTSSintheZnSeterracehasasetcapacitanceCZnSe.(b)TheelectricbetweenthetipandtheTSSintheBi2Te3layerwithacapacitanceofCBi2Te3.Herewewouldseeasmallercapacitancebecauseofthelargerdistancebetweenthetwocapacitive\plates"..............................91FigureB.6:CompressibilitymeasurementtakenovervariousbareBi2Te3at77K.Whilethereisanotableslopepresentinthesemeasurements,weseewhatwebelievetobetheDiracconenear1{1.5Vinthesecurves.ThiswouldthekeysignatureofthetopologicalsurfacestateandisexpectedwhenoverabareTI.Allcurvesshownareaveragesofthousandsofindividualcurvesinordertosuppressnoiseandenhancethesignal.Theblackcurvesaretherawdatawhiletheredcurvesaretheresultsofsmoothingtherawdata....................................93xviiFigureB.7:CompressibilitymeasurementtakenovervariousZnSeterracesat77K.Asidefromtheapparentslope,therearenonotablefeaturesinthesemeasurements.ThelackofastepinthecapacitanceorappearanceoftheDiracconeindicatesthatthereisnoinducedtopologicalsurfacestateinthisspinterface.Allcurvesshownareaveragesofthousandsofindividualcurvesinordertosuppressnoiseandenhancethesignal...94FigureB.8:(a)STMtopographshowingthecleartregionsofZnSeandBi2Te3.(b)STMspectroscopymeasurementtakenonbareBi2Te3showingtheusualdensityofstatesforaDiracconeatroomtemperature.......95FigureB.9:(a)Kelvinprobetopographshowingregionswithsurfacepo-tentials.PleaserefertoAppendixCforadiscussiononhowthenullingvoltageVNisacquired.(b)Kelvinprobespectrumtakeninthedarkregionatthebottomof(a).TheresultingnullingvoltageVNis-1.035V.(c)Kelvinprobespectrumtakeninthebrightregionof(a).Here,VNis-1.132V.WhileitwouldhavebeeninterestingtotakeSTMspectroscopyintheseregionstoseeiftheDiracconeshifted,itwasnotpossiblewiththissampleduetopoorconductivityat77K...............96FigureC.1:AplotsimulatingthreedC=dzmeasurementstakenatthesamepointattvaluesofz0.Herewecanseethecrossingpointoccursatavoltageof.5Vandatameasuredvalueof4fordC=dz.Thistellsusthatthenullingvoltagewouldthenbe.5Vandthearbitraryfromthemeasurementelectronicshasavalueof4..................100xviiiChapter1IntroductionScanningtunnelingmicroscopy(STM)hasbecomeastapleformeasuringtheelectronicstatesinconductingmaterialsincondensedmatterphysics.Byutilizingoneofthemanyuniquephenomenaofquantummechanics,knownasquantumtunneling,aconductingtipcanbebroughtintocloseproximitywithavoltage-biasedconductingmaterialandanelectricalcurrentwillw.Thisoccursbecauseindividualelectronstunnelthroughthevacuumbarrierbetweenthetipandthesample.Thismeasurementtechniqueallowsustonotonlymapoutthetopographyofasample,butalsotoprobethelocaldensityofstatesofthematerialaswell.Bythepointoveralocationofinterestandsweepingthebiasvoltage,wecanmeasurethecurrentresponseandextractthedensityofstates.Thisisanexceptionallypowerfulandtoolwhenitcomestoobservingthedetailsoftheelectronicstatesnearinterfaceswithtmaterialsplacedinproximitytooneanother.Inthisdissertation,Iwillfocusontheinterfacialinteractionsofsuperconductorsandtopologicalinsulatorsaswellasconventionalinsulatorsandtopologicalinsulators.Three-dimensionaltopologicalinsulatorsarearecentlydiscoveredclassofsemiconductorforwhichintrinsicelectronspin-orbitcouplinggivesrisetospinpolarizedchannels;thesechannelscrossthesemiconductinggapwithalineardispersionrelation,whichiscommonlyknownastheDiraccone.StatesfoundintheDiracconeresideonsurfacesandhavetheirspinandmomentumcoupledinveryunusualways.Thesestatesareknownastopological1surfacestates(TSS),andwillbeoneofthecentralfocusesoftheelectronicmeasurementspresentedinthisdissertation.Anothercentralfocuswillbeinthephenomenonknownassuperconductivity.Supercon-ductivitywasinitiallydiscoveredasaphaseofmatterinwhichthereiszeroresistancebelowsomecriticaltemperature.In1957,Bardeen,Cooper,andScprovidedthemicroscopictheorytodescribethisphenomenon,whichisnowknownasBCStheory.Superconductivityarisesfromcompletelyerentphysicsthantopologicalinsulators:aspontaneously-brokengaugesymmetryresultsinCooperpairing,whichisanintricatedanceofelectronchargeandspin.[1]Thisiscrucialtosuperconductivityasbosonsarecapableofinhabitingthesamequantumstate,unlikefermions.Furthermore,Cooperpairsarecapableofmovingfromonematerialtoanother.Whenasuperconductorisplacedincontactwithanormalmetal,theCooperpairsdojustthat,andthenormalmetalexperiencesinducedsuperconductivityouttoadistance.Thisisknownasthesuperconductingproximityandisoneoftheprimaryfocusesofthisdissertation:specthesuperconductingproximityinthetopologicalsurfacestateofatopologicalinsulator.Theinducedsuperconductivityinthetopologicalsurfacestateispredictedtoformatwo-dimensionalp-wavesuperconductor,andisofgreatinteresttothesciencommunitybecausethisspformofsuperconductivityisbelievedtohouseMajoranafermions,whichareparticlesthatarealsotheirownantiparticles.Agreatdealofworkwentintostudyingtheinterfacialofconventionalinsulatorsontopologicalinsulators,whichispresentedintheAppendixB.Ithasbeenpredictedthatthelocationofthetopologicalsurfacestatecanbetunedbyplacingaverythingconventionalinsulator(CI)ontopofatopologicalinsulator.[2]Thetopologicalinsulatorcaneither\topologize"theCI,causingtheTSStotothesurfaceoftheCIandawayfromthetoplayeroftheTI.Likewise,theCIcouldtrivializethetoplayeroftheTI,pushingtheTSS2downonelayerintotheTI.ThethirdoptionisthetraditionallyexpectedbehaviorinwhichtheTSSremainsattheTI/CIinterface.Thesethreepossibleareknownasthe\dual"proximityForthesemeasurementsweutilizeaspecialscanningprobemethodknownassubsurfacechargeaccumulationimaging(SCAI).Thisdissertationwillpresentalloftheabovematerialasfollows:IwillintroducetherelevantscanningprobemethodsusedinthisworkinChapter2.Next,inChapter3,Ipresenttherelevanttheoriesnecessarytohaveabasicunderstandingoftheresultspresented,namelythetheoriesoftopologicalinsulatorsandsuperconductivity.InChapter4,IshowtheresultsofSTMmeasurementstakenontopologicalinsulatorswithsuperconductingislandsdepositedonthesurface,whichdemonstratesomestrikingbehaviorsinthedensityofstates.Finally,IconcludethisdissertationwithadiscussionofmyresultsandthefutureexperimentsthatmyworkhaslaidthepathforinChapter5.Adiscussiononpreliminaryworkinthesuperconductingproximityandthedualproximityofconventionalinsulator/topologicalinsulatorinterfacescanbefoundintheappendices.3Chapter2ExperimentalTechniques2.1ScanningTunnelingMicroscopyScanningTunnelingMicroscopy(STM)isaverypowerfulandwellestablishedmeasurementtechniqueintheworldofsurfacephysics.Byplacingananoscaleconductingtipinverycloseproximitytoaconductingsample(ontheorderof1nmaway)andapplyingavoltagebetweenthetwo,wecanimagestructuresonthescaleofangstromsthroughtheuseofthequantummechanicalphenomenonknownastunneling.ButthetrulypowerfulcapabilitiesofSTMcomefromtheabilitytotakespectra,whichallowustomeasurethelocaldensityofstates(LDOS)oftheconductingsample.WhileSTMmayappeartobeaverysimplemeasurementmethod,theunderlyingphysicalmechanismsandexecutionsareactuallyverycomplex.2.1.1TunnelingTheorySTMisdirectlydependentonquantummechanics,moresponelectrontunneling.Thismechanismcanbesimpdowntoaone-dimensional,trapezoidalbarrierofwidthlinbetweentworegionsofFermienergy.Thebarrierisformedbythetworkfunctionsofthematerials.TakingtheleftsideofFig.2.1tobethesampleandtherightsidetobethetip,itcanbeseenthatanincomingelectronfromthesample'ssidecanactually4tunnelacrossthisgapintothetip.Thisisgovernedbythebehavioroftheelectronwavefunction,z),anditsinteractionwiththebarrier.Thewavefunctioniswellunderstoodtoundergoexponentialdecayinsidethebarrier,takingtheformz)=Ae,whereAisanormalizationfactor.Assumingthebiasenergy,eV,issmallcomparedtotheworkfunctionsofthesampleandtip˚Sand˚T,wecanapproximatethebarrierasrectangularandthebarrierheightwouldbe˚avg=(˚S+˚T)=2,andtheexponentialdampingfactoris=p2m˚avg=~.Aftertunnelingthroughthebarrier,thewavefunctionresumesitsFigure2.1:Vacuumbarrierbetweenatipandsample,withtheelectronwavefunctionshowninred.Thisdemonstratesthatapropagatingelectronhastheabilitytotunnelfromthesampletothetipwhenavoltageisappliedbetweenthem.Here,eVisthecontributionfromthebiasvoltage.oscillatoryformz)=eikFz,whereisthetransmissioncotandkFistheFermiwavenumber.Becausethewavefunctionmustbecontinuousatthebarrierinterface,wecanclearlyseethatl)=Ae=jjwhentheslopeofinsidethebarrierissmall;thewavefunctioninsidethebarriermustequaltheamplitudeinsidethetipattheinterface.Duetothefactthatthetunnelingcurrentisproportionaltothetotransmissionprobabilityjj2,5wearriveattheimportantrelationI/e2(2.1)Thisrelationishighlyovduetootherfactorssuchasthethree-dimensionaltipgeometryandthermallyexcitedelectronicstatesinthetipandsample,butthecrucialpointisthatthetunnelingcurrenthasanexponentialdependenceonthedistancebetweenthetipandsample.Sincetheworkfunctionofregularmetalsisaround5eV,ˇ1A1,implyingthatifthetipmovesonly1Afurtherfromthesamplethetunnelingcurrentwoulddrop1orderofmagnitude.ItisduetothisexceptionalsensitivitythatSTMissuchaneandsensitivetoolwhenscanninginthexy-plane.2.1.2ScanningTunnelingSpectroscopyArguablythemostpowerfulaspectofSTM,ScanningTunnelingSpectroscopy(STS)allowsforthemeasurementofthesample'sLDOS.Bythetipoveradesignatedlocationandsweepingthebiasvoltage(seeFig.2.2),weareabletoacquirethesample'sI-Vcharacteristics,andmoreimportantlyitstialconductancedI/dV.ThisquantityisrelatedtotheLDOS,butitisbythermalexcitations(ontheorderof25meVatroomtemperature);STSisexceptionallyeattemperaturesnear0K,wherethermalenergyisnearlygone.TofullyunderstandthesigofSTS,wewillnowdiscussqualitativelyhowdI/dVrelatestothesample'sLDOS.Wemustassumethatthetunnelingtransitionoccursatconstantenergyoncethebiasvoltagehasbeenapplied.Wecanthensumallofthecontributionsfromeachenergyleveltocalculatethetotaltunnelingcurrent.SinceweareworkingatatemperatureT>0,thenumberofoccupiedstatesforasampleissimply6Figure2.2:Bysweepingthebiasvoltage,weareableto(empty)theempty(occupied)statesofthesample.Asthetunnelingcurrentdependsonthenumberofaccessiblestates,thisallowsustoextractitsLDOS.thesample'sdensityofstates,Ns(E)convolvedwiththeFermifunctionf(E).Here,theFermifunctionisasf(E)=1=(1+eE=(kBT)),andtheoccupiednumberofstatesissimplyasNs(E)f(E).Electronsfoundinthesestatescanonlytunnelintotheemptystatesfoundinthetip.ThisvalueisgivenbyNt(EeV)(1f(EeV)).Puttingitalltogether,wecanseethatthetunnelingcurrentfromsampletotiptakestheformIs!t/Z+1jj2Ns(E)f(E)Nt(EeV)[1f(EeV)]dE(2.2)wherejj2isthetunnelingprobability.AssumingthetransmissionprobabilityisindependentofEforenergiesclosetotheFermilevel,itcanbepulledoutoftheintegralandfactoredintoaprefactorinourproportionalityequation.Toobtainthenettunnelingcurrent,wesimplyneedtosubtractthecurrentgoing7fromthetiptothesample,It!s.ThisresultsintheequationI/Z+1Ns(E)Nt(EeV)[f(E)f(EeV)]dE:(2.3)TosimplifyEq.2.3further,thefactthatourSTMtipsareusuallymadefromnoblemetalsandalloys,wecanassumeNt(E)isapproximatelyconstantneartheFermilevel,soNtisindependentofE.IfweerentiatethetunnelingcurrentinEq.2.3withrespecttovoltage,eV,wegetdIdV(V)/Z+1Ns(E)@f(EeV)@(eV)dE(2.4)Equation2.4canbeevenfurtherduetothefactthat@f(EeV)=@(eV)isabell-curvecenteredatE=eVwithawidthontheorderofkBTandunitaryareabeneaththecurve.Thus,askBT!0,equation2.4downtodIdV(V)T=0/Ns(eV)(2.5)anditisdirectlyevidentthatatlowtemperatures,thedtialconductancedI=dVisdirectlyproportionaltothesample'sdensityofstates.STSisanexceptionallypowerfultoolduetothefactthatallofthiscanbedoneonamobiletip,allowingustomeasurethelDOSanywherewewishonthesamplesurface.82.1.3ExperimentalSetup2.1.3.1BesockeDesignSTMThemicroscopesusedinourlabarehouse-builtBesockedesignSTM's.TheBesockedesignconsistsoffourpiezoelectrictubes:threeofwhichholdandmanipulatethesampleandonetorastertheSTMtipoverthesurface.[3]AschematicofthissetupcanbeseeninFig.2.3(a),andtherealmicroscopecanbeseeninFig.2.3(b).BesockedesignSTM'sutilizeanFigure2.3:(a)SchematicshowingthelayoutofaBesockedesignSTM.(b)Actualhouse-builtSTMintheTessmerlab.inertialapproachmethodforbringthesampleintotunnelingrangewiththetip.Withtheconductinghemispheresontopofthecarrierpiezotubessupplyingthevoltagetothesample,9thecarrierpiezotubesbendinasuchamannertobringthesampleholderclosertothetipthenabruptlysnapback.Duetothelargeinertiaofthesampleholderandlowfrictionbetweentheconductinghemispheresandthesampleholder,thesampledoesnotmovebacktoitsinitialposition.Whilethisishappening,thetippiezotubeextendsinanattempttoatunnelingcurrent.Iftunnelingcurrentisnotproduced,thetipretractsandtheentireprocessisrepeateduntilthesampleisintunnelingrange.[3]ThisdesignisidealforcryogenicSTMmeasurementsduetotherigidityofthesetup.Becauseallofthepiezoelectrictubesaresolderedtothesamebaseplateandareidenticaldimensionally,thermalexpansionandcontractionarenolongermajorissueswhenscanningatlowtemperature.2.1.3.2CryogenicSTMSystemOurSTMsareplacedontheendofalongdippingstickinorderforustobeabletolowerthetip-samplesystemintoacryogenicdewar.Inourlabweusetwoliquidheliumcryostatscapableofreachingatemperatureof300mK,whichutilizesuspensionmethodsinordertoisolatethewholesystemfromvibrations.ThecompletesystemwithmicroscopefullyloadedintothedewarisshowninFig.2.4.Thecryostatconsistsofacentralsampletubeintowhichthemicroscopeisloweredfromthetop.Thissampletubeistypicallykeptatavacuumofaround5106torr,andisidealforkeepingthesamplesurfacecleanandfreeofcontaminants.Thissampletubealsocontainsacarbonbasedsorptionpump(sorb)toabsorbanyexcessgasesinthechamber.Outsideofthiscentralchamberisaninnervacuumchamber(IVC)whichservestoisolatethewallsofthesamplechamberfromthewallsoftheheliumbath.Inordertocoolthemicroscopemoreely,wetypicallyhaveasmallamountofheliumexchangegasinthe10Figure2.4:ThefullmeasurementsetupwithSTMdipperfullysubmergedintheblueliquidHeliumdewar.11IVCtoequilibratethewalls.OutsideoftheIVC,wehavethehelium/nitrogenbath,wherethecryogenicliquidisstoredforthedurationofourcryogenicexperiments.Finally,wehavetheoutervacuumchamber(OVC),whichservestoisolatethecryogenicbathfromoutermostwallsofthecryostatwhichareatroomtemperature.TheOVCalsohousesmultiplelayersofmylar,whichshieldthemicroscopefromoutsideradiationsources.AsimplediagramofthecryostatcanbeseeninFig.2.5.Insideofthiscryostataretwomoreimportantdevices.First,wehaveasuperconductingmagnetthatcansupply5or10Tdependingonwhichsystemweuse.Theotherimportantdeviceinthedewaristhe1Kpot,whichallowsustocoolthemicroscopedownto300mKwhenusing3He.Thisisalsocrucialbecausesomesignalsthatwemeasurearestronglysuppressedduetothermalexcitations.Whenthe1Kpotisnotinuse,thesystemisusuallyoperatingatatemperatureof4.2Kfromthe4Hebath.2.2ScanningChargeAccumulationImagingWhileSTMmaybeanexcellentprobeofconductingmaterials,itstillhasonemajordrawback.Manytimesthereisagreatdealofelectronicstructureoccurringmanynanometersbeneaththesurface.ItisinthisregimethatSTMfallsshort.Fortunately,anotherscanningprobemethodwasdevelopedbyDr.StuartTessmerandDr.RayAshoorithatallowsforprobingthesesubsurfacestates,knowasscanningchargeaccumulationimaging(SCAI).AsidefromSTM,SCAIistheTessmerlab'smainareaofexpertise.[4,5]ThismeasurementmethodisespeciallyrelevantforthedatapresentedinAppendixB.SCAIishighlysensitivetocharges,withanoiselevelof:03e=pHzat0.3K.Becausethismethodissuchafantasticdetectorofchangesincharge,itisprimarilyusedtomeasure12Figure2.5:Diagramofcryostatusedinourlab,showingalloftheimportantchambersandfeatures.Here,thesorptionpump(Sorb)isintheshapeofatorusandiffoundinsideofthevacuumchamber,whichallowstheSTMtopassthroughitandreachthebottom.13capacitancebetweenthetipandthesample.Thiscanbedonebysweepingthebias-voltagebetweenthetipandthesamplewhilemeasuringthechargeaccumulationonthetip,whichunderidealconditionsperfectlymirrorsthechargeaccumulationinthesample.ThecapacitanceisthenextractedfromthesequantitiesbytheequationC=dqdVuqV(2.6)whereqistheoscillatingchargesignaldetectedbythelock-in,andVistheACexcitationvoltage.FromhereonoutweshalllabelthisquantityasthemeasuredcapacitanceCm.LikeSTM,SCAIiscapableofmeasuringthedensityofstatesofthesample,butinsteadofthelocaldensityofstates,SCAImeasuresthethermodynamicdensityofstates(DOSTh).Inthesemeasurements,DOSThcanbeextractedCmusingthefollowingrelation:ACm=ACG+1edn(2.7)Here,weestimatethegeometryofthemeasurementasaparallelplatecapacitorofareaA,whereCGisthegeometriccapacitancebetweenthetipandsampleandistheDOSTh.ThegeneralofCGisshowninFig.2.6.BecauseweusethesamematerialforourtipinSCAIaswedoforSTM,wecanassumethatthedensityofstatesforthetipisconstant,aswedidbeforeinSection2.1.2.SinceCmisalwaysmeasuredwithastationarytip,CGwillremainconstantandanychangesasafunctionofDCbiasvoltageseeninthecapacitancesignalwillstemfromtheDOSTh.Whilethismeasurementmayseemstraightforward,anintricatecircuitisrequiredinordertofullresolvethesecapacitivechanges,whicharenormally14Figure2.6:Ahighlydiagramdemonstratingtherelationshipofgeometriccapaci-tanceCGonthedistancebetweenthetipandthesample.onthescaleof1aF,orattoFarad(1018F).ThemostimportantcomponentinourSCAIcircuit(asidefromthescanningprobetip)isthehigh-electron-mobilitytransistor(HEMT),whichiswhatgivesussuchincrediblechargesensitivity.ThescanningprobetipisconnecteddirectlytothegateoftheHEMT,whichallowsustomeasurevoltagechangeinthetip.TheHEMTcircuitgivesusexcellentchargesensitivityduetothesmallcapacitancebetweenthegateandthesource/drainchannels,asdiscussedbelow.AdiagramoftheHEMTcircuitusedinthesemeasurementsisshowninFig.2.7(a).Inordertogetthemaximumsensitivitytothesignalofinteresttothelock-inwehaveplacedastandardcapacitor(Cs)inthecircuit.Cssubtractsawaythebackgroundcapacitance,stemmingfromelectricthatdonotterminateonthetipapex.ThisisaccomplishedbysupplingCswithanACvoltagethatisofequalamplitude,Vexc,but180outofphasewiththeACvoltagesuppliedtothesample.Fromhere,wecanmodeltheremainderofthecircuitasasimplevoltagedivider,asshowninFig.2.7(b).Indoingso,we15Figure2.7:(a)CircuitdiagramofourSCAImeasurementchip.TheHEMTactsasahighlysensitivechargedetectorthatisdirectlyconnectedtothescanningprobetip,whichallowsustoseechargeuctuationsinthesampleduetoimagechargesformedinthetip.(b)eddiagramdemonstratingtheexistenceofavoltagedividerfortheACexcitationvoltagegoingthroughtwocapacitors.16mustconsiderVinandVout,whichpertaintoVexcandVm=Grespectively,whereGisthegainfactorforthecircuitoforderunity.Inordertoconstructthisvoltagedividerequation,wemustrsttheintrinsiccapacitancesthatmustbeaccountedforintheassemblyofthecircuit.Cinistheinputcapacitance,whichisthecapacitancebetweenthegateandthesource/drainchanneloftheHEMT.Cinistypically0.3pF(picofaradsor1015F),aroundthreeordersofmagnitudelowerthanCm.Again,Cmisthetip-samplecapacitance,whichisproportionaltothechargeonthetip.ThisresultsinthevoltagedividerequationVm=GVexcZ2Z1+Z2(2.8)whereZ1=1=i!CmeasandZ2=1=i!Cin.BecauseCinis3ordersofmagnitudelargerthanCm,wecaneliminateZ2inthedenominatorofEq.2.8.Thus,theresultingvoltagemeasuredbythelock-in,Vm,isVm=GVexcCmCin(2.9)FromEq.2.9,itisclearthatthesmallerCinis,thelargerVmwillbe.BecausestraycapacitancefromthecircuitalsocontributestoCin,wemustbeverycarefultomakethemeasurementcircuitasphysicallysmallaspossible,usuallyplacingtheHEMTwithin1mmofthetip.AfullyassembledmeasurementchipandapictureoftheHEMTcanbeseeninFig.2.8.17Figure2.8:TheleftimageshowsafullyassembledSCAImeasurementchipcompletewithprotectionloops.ThemethodofassemblyutilizestweezerworkwithconductiveepoxytofastentheprimarycircuitelementsandconnectionwirestotheGaAschip.Inordertoconnectthegoldpadstothecorrectelements,weutilizeagoldwirebonder.TherightimageshowsaHEMTwithnoconnections.TheHEMTasawholeisapproximately600mwideandthegatepaditselfisonly80mwide.182.3TipPreparationForourscanningprobemeasurements,weusePtIrtipsduetotheirhighlyinertcharacteristicsandidealconductivity.Inordertoobtainatomicresolutionimagesandhighqualityspectra,itisimportantthatthetiphaveastable,atomicallysharppointontheend,orattheveryleasthaveaverysharpradiusofcurvaturethatitnominallylessthana30nm.Typically,thewayweformthesesharptipsisthroughmechanicalcutting,butthereisusuallyonlya50%successrateofgettingtlystabletipsusingthismethod.TheusualgeometryofcuttipscanbeseeninFig.2.11.Inordertoimprovethesuccessrateoftipfabrication,wehaveinvestigatedafewmethodsoftipetching,inwhichthetipissubmergedinaCaClsaltsolutionandavoltageisappliedbetweenthetipandacathode.TheinitialstepsofthisetchingprocesstypicallyformtipswithashapeshowninFig.2.9,wherealargebulbofmaterialisleftontheendofthetip.Thisbulbistypicallynotidealforscanningprobemeasurements,sothenextstageofetchingisknownasetching.Here,thetipisputinathinoftheCaClsolutioncapturedinasmalltungstenhoop.Asmallervoltageisappliedbetweenthetipandthehoop,andtheneckoftheetchedtipisgraduallyetchedawayuntilthebulbdropsWhenthebulbdropsasharp,symmetricalpointremainsontheendofthetip,usuallywithofradiusofaround20nm.Thisgiveusanidealtipforallofourmeasurementmethods,fromSTMtoSCAI.Unfortunately,whentipshavearadiusofcurvaturethissmall,theybecomeveryfragileandsensitivetoanyphysicalcontact.InFig.2.10,weshowanetchedtipthatwasdroppedwhenbeingmountedintotheSEMforcharacterization.Thistipisnolongerusableinthescanningprobeduetothemajorradiusofcurvatureatthebendingpoint.Thisjustgoestoshowthatanyminorcontactontheendofthetipcangreatlyjeopardizetheintegrityofthe19Figure2.9:SEMimagesdemonstratingthetipgeometryaftertheroughetchingstep.Thebulbattheendwillideallydropduetogravityassistanceduringtheetchingprocess.(a)Initialscanwherethebulbisclearlyvisibleand(b)Closerlookatthesametip.experiment,especiallyifthetipcomesincontactwiththesamplein-situ.Figure2.10:SEMimagesdemonstratingwhathappenstoanSTMtipwhenitisdropped.Whatwasonceanidealcandidateforuseasascanningprobeisnowrenderedunusableduetothelargeradiusofcurvatureatthebendinthewire.(a)Initialscanwherethebendisclearlyvisible.(b)Closerlookatthesametip.Afteragreatdealoftryingtoarepeatablemethodfortipetching,weeventuallyfoundthatthemostproductivewaytomaketipsistocut8-10tipsmechanicallyandimagetheunderSEMforcharacterization.Usingthismethod,wenotonlyhaveachanceofproducingtipsthatareinfactsharperthanthosemadebychemicaletching,butalsoknow20Figure2.11:SEMimagesshowingtheofasinglecuttip.(a)Whilethecuttiplookstohavemultiplepoints,thereisoneclearpointthatismorethan50furtheroutfromtheothers.(b)Closeupoftheapexshowingthattheradiusofcurvatureislessthan40nm.thetipgeometry,whichcanbeveryimportantforsomeoftheexperimentsdoneinthisdissertation.AnexampleofanidealSTM/SCAItipisshowninFig.2.11.Herethetiphasaverysharpradiusofcurvature,whichisbothidealfortunnelingandcapacitancemeasurements.Whilemechanicallycuttipsoftenhavemultiplepointsattheend,allthatmattersisthatasinglepointsticksouttlyfurtherthantheothers.Thisensuresthatweknowwhichpartofthetipwillbeusedforthemeasurements.InthecaseofthetipshowninFig.2.11,itiscleartoseethatthereisaprominentpointthatsticksoutfurtherthantheothers,whichwhenloadedintothescanningprobesystem,wouldbethetunneling/\capacitiveplate"pointinthemeasurements.21Chapter3TheTheoriesofTopologicalInsulatorsandSuperconductivityTherearetwoprimarytheoriesattheheartofthisthesis:superconductivityandtopologicalinsulators.Inthischapter,wewilldiscussthemainfeaturesofthesetwogenerallycomplicatedtheoriesinordertogiveabasicunderstandingoftheimportantphysicsobservedinthisdissertation.3.1TopologicalInsulatorsTopologicalInsulators(TI)arearelativelyrecentdiscoveryintheworldofcondensedmatterphysics.Topologicalinsulatorsarematerialsthatareinsulatingintheinterior(asthenameimplies).ThismeansthattheFermilevelliesbetweenelectronicbands;thebandbelowcannotconductelectricityaseverystatethatisoccupiedthatwouldcarrychargeinonedirectionhasatwinstatecorrespondingtotheoppositedirection,ensuringthatthenetmomentumiszero.TounderstandthebasicprinciplesofTIs,Iwillintroducesomemorebasicprincipleswhichwillhelpclarifytheunderlyingmechanicsthatareimportanttotopologicalinsulators.First,wewilldiscussaquantitycalledtheCherninvariant,whichcanbeusedtodescribethetopologicalnatureofamaterial.Then,wewilldiscussthequantumHallandquantumspinHallinSec.3.1.1Itisimportanttonote22thatwhilethetopicofthisdissertationisnotonquantumHallphysics,thesequantumHallsystemsdisplayphysicswhichisanalogoustothethree-dimensionalmaterialsIstudied,andareusefultointroducethenatureoftopologicalinsulators.Sec.3.1.2presentsamorerigorousintroductiontothree-dimensionalTIs.TheChernnumberisanimportantquantityinTIsystems.ItcanbecalculatedbyinvestigatingtheBerryphase,whichisageometricphasethatasystemacquireswhenitundergoesaclosed,adiabaticcycle.TheBerryphasecanbephysicallyunderstoodbylookingatthebehaviorofBlochwavefunctionsjum(k)iwhentransportingkaroundaclosedloop.TheBerryphaseisthegivenbythelineintegraloftheBerryconnectionAm=ihumjrkjumi.TheBerryphasecanalsoexpressedasthesurfaceintegraloftheBerryFm=rAm.TheCherninvariantissimplythetotalBerryintheBrillouinzone,calculatedbytheequationm=12ˇZd2kFm(3.1)wherekistheelectronwavenumber.ThetotalChernnumberisthesumoveralloccupiedbands,=PNm=1m,whichturnsouttobeaninteger.[6]ThisChernnumbercanbeusedtodenotethetopologicalnatureofamaterial.VacuumandothertrivialinsulatorshaveaChernnumber=0.Notopologicallyinterestingphysicsoccursacrosstheinterfaceoftwomaterialswith=0.TheChernnumberistopologicalinthesensethatitisinvariantundersmalldeformationsoftheHamilton.Inotherwords,becausetheChernnumbermustbeaninteger,itcannotbechangedatallunderdeformationsthathaveenergieslessthantheenergybetweenadjacentm's.However,largedeformationsoftheHamiltoniancancausethegroundstatetocrossoverotherbands.ThismustoccuratinterfaceswheretheChernnumberchangesfromoneintegertoanother,givingriseto23partiallyconductingstatesneartheboundary.NothinginterestinghappensattheinterfaceoftrivialinsulatorsastheChernnumberdoesnotchangecrossingintothevacuum.ThematerialsofinteresthaveChernnumbersotherthanzerosothatinterfaceswiththevacuumwheretheChernnumbermustchangeproduceveryinterestingTogiveaconcreteexample,belowwewillnowdiscussthequantumHall(Sec.3.1.2presentsarigorousintroductiontothree-dimensionaltopologicalinsulators).3.1.1QuantumHallandQuantumSpinHallInordertofullyunderstandthedrivingphysicsinTIs,ithelpstoinvestigatethequantumHallIfwelookatatwo-dimensionalelectronsystemandapplyalargeenoughmagneticnormaltotheplaneofthissystem,theHallconductanceofthematerialcanbeexplainedbytheequation˙H=e2=h(3.2)where˙HistheHallconductance,eistheelectroncharge,hisPlanck'sconstant,andisanintegerknownasthefactor.[7]TheHallconductanceisquantizedinunitsofe2=h.Theappliedtothistwo-dimensionalsystemcausestheelectronicstatestoformintoLandaulevels.ThefactoristhenumberofLandaulevelsandisby=nhc=eB.[8]ItturnsoutthatthefactoractuallyistheChernnumberforthiscase.Letssupposethat=1intheinterior;astheLandaulevelrepresentsaband,theinterioristhenaninsulator.Attheedge,asweconsidercrossingtheboundarytovacuum,thedensityhastogotozero.ThisnecessarilymeansthattheLandaulevelcannotbecompletelyattheedge.Henceaconductingstatemustoccur.Generallyspeaking,whentheChernnumberchangesfromoneintegertoanother,theremustbeapartially24conductingstatesneartheboundary.ThegeneralbehaviorfortheexampleabovecanbeseeninFig.3.1.Theelectronsontheedgeofthetwo-dimensionalelectronplanetravelalongtheedgeofthematerial,whiletheelectronsinthecenteraretoorbits.Thesetwoelectronstatesrepresentconductingandinsulatingstatesrespectively.Moreover,theseedgestatesareactuallyimpervioustonon-magneticdefectsinthematerialduetothisquantization,andhaveallowedforthediscoveryofastandardvalueforelectricalresistanceknowasthevonKlitzingconstant,RK=h=e2=25;812:807557ThequantumHallsystemcanalsobeusedtocalculatethestructureconstanttoveryhighprecision.[9]Figure3.1:AdiagramofthequantumHallsystemwithintegerLandaulevelsintheinteriorintheclassicalpicture(a)andtheresultingquantumbehavior(b).Theedgeofthetwo-dimensionalsystemconductswhilethecenterremainsinsulatingduetothelocalizedstatesoftheelectrons.Figureadaptedfrom[6].Morerecently,interestwastakeninapossiblewaytocreatetheseedgestateswithouthavinganappliedmagneticthroughthetwo-dimensionalsystem.ThisresultedinthediscoveryofthequantumspinHall(QSHE)byKaneandMele.[10]ThequantumspinHalldoesnotneedanexternalmagneticduetothefactthatthereisanintrinsictraitknownasspinorbitcoupling,whichdoesnotbreaktimereversalinvariance.[10,11]25Thisresultsintwospinpolarizededgestatesinourtwo-dimensionalsystem,asshowninFig.3.2.TheresultingofthisspinorbitcouplingcreateswhatisknownastheDiraccone,asshowninFig.3.3.TheDiracconestatesconnecttheconductionbandtothevalencebandthroughthesetwocrossingchannels.ThenoveltopologicalsurfacestatesoftopologicalinsulatorsexistsolelyintheDiraccone,whichisfoundinsidethebandgap.Otherwise,theremainingbulkofthematerialisideallyinsulating.Inthiscase,thetotaledgecurrentendsupbeing0,buttheindividualspinpolarizedcurrentsattheedgehaveunitaryconductance˙s=e2=2h.[11]Thematerialfoundtoexhibittheintrinsicbehaviorwastwo-dimensionalgraphene,whereKaneandMelewereabletodemonstratetheexistenceoftheQSHEusingthetightbindingmodel.[12]ManyresearchersnowconsiderthequantumHallmaterialstobethetopologicalinsulators,althoughonlytwo-dimensional.Inordertomoveforward,wemustexpandourthinkingfromtwo-dimensionstothree,asthetopologicalinsulatorsusedinthisdissertationarethree-dimensional.Figure3.2:Thisdiagramdemonstratesthetwospinpolarizedconductingstatesontheedgeofthetwo-dimensionalsystem,wheretheredlinecanbeconsideredtobespin-upelectronsandthebluecanbeconsideredtobespin-downelectrons.Figureadaptedfrom[6].26Figure3.3:TheDiracconeisaconsequenceofthespin-orbitcouplingfrominTI's.Theredandbluecrossingchannelscorrespondtothespin-upandspin-downchannelsrespectively.ThecrossingpointinthecenterisknownastheDiracpoint,andistypicallynotfoundattheFermilevelEFofthematerial.Figureadaptedfrom[6].3.1.23DTopologicalInsulatorsandBi2Se3Three-dimensionaltopologicalinsulatorsareaspecialsetofmaterialsinwhichaboundarywithatrivialinsulator,suchasvacuum,allowsforthecreationoftopologicallyprotectedsurfacestates.Thistopologicalprotectioncanbedescribedbythenotionthattime-reversalinvariantperturbationscannotopenagapinthetopologicalsurfacestates,butperturbationsthatbreakTRsymmetry,suchasanexternalmagneticcanopenagapinthesesurfacestates.ThesematerialscanbedescribedbywhatisknownastheChernparity.Athree-dimensionalTIhasfourChernparities,deby0,1,2,and3,whichcanallbecalculatedfromthebandstructureofthematerial.TheChernparities13pertaintothexy,xz,andyzplanesofthematerial,while0whetherornotthematerialisastrongtopologicalinsulator,aweaktopologicalinsulator,oratrivialinsulator.[13]IfthereareanevennumberofDiracpointsenclosedbytheFermisurface,theChernparity=0,27whichisacharacteristicofatrivialinsulatororaTIwithweakspinorbitcoupling.ForanoddnumberofenclosedDiracpoints,=1,andthematerialisaTIwithstrongspinorbitcoupling.[14]ItisonlyatinterfaceswheretheChernparitygoesfrom=1to=0(muchliketheChernnumberforaquantumHallsystem)thatthetopologicalsurfacestateexists.Sincethetheoreticalformulationanddiscoveryoftopologicalinsulators,manymaterialshavebeendiscoveredthatcanbeastopologicalinsulators.Thetruethree-dimensionaltopologicalinsulatorfoundwasBiSb.[15]FollowinginthefootstepsofBiSb,morecommonlyusedmaterialswereshowntobeTIs,suchasBi2Se3,Bi2Te3,andSb2Te3.[16]Forthethree-dimensionalTIBi2Se3,thecrystalcanbedescribedbylayersconsistingof2Biatomsand3Seatoms,knownasquintuplelayers(QL),asshowninFig3.4(a).ThechemicalbondoftheBiandSeatomsintheseQLsisverystrong,whilethebondingstrengthbetweenindividualQLsisweakandcanbedescribedbyvanderWaalsattractions.Thisconvenientlyallowsustocleavethecrystalsbyusingscotchtapeandpeelingthetopfewquintuplelayersfromthecrystal,leavingaclean,atomicallysurface.ForBi2Se3,theexistenceofthetopologicalsurfacestatescanbedescribedbyamodelwhichincorporatesthestemmingfromthechemicalbondingofBiandSeinthequintuplelayersofthecrystal,thecrystalofthematerial,andthebasicspinorbitcouplingintrinsictothesematerials.[17]ThestatesnearesttotheFermienergyexistintheelectronp-orbitalsoftheBiandSe,soweonlyneedtoinvestigatethoseorbitalstosuccessfullydescribethecreationofthetopologicalsurfacestate.Theenergystatesofthesep-orbitalsarelabeledasP1x;y;z,P2x;y;z,andP0x;y;z,wheredenotestheparityoftheenergystate;thebehavioroftheenergystatesisshowninFig.3.4(b).ThechemicalbondbetweentheBiandSeatomshybridizestheirenergystates,distancingthemfromtheFermilevel,asshowninFig.3.4(b)region(I).Fig.3.4(b)region(II)demonstratesthe28ofthecrystalsplittingofthematerial.ThepzstatesforboththeBiandSeatomsarebothsplitfromtheirrespectivepx;ycomponents,anditturnsoutthattheP1+zandP2zstatesbecometheclosestenergylevelstotheFermilevelwhiletheirrespectivepx;ystatesremaindegeneratefurtherfromtheFermilevel.Finally,wecanincorporatethespinorbitcouplingofthematerialbyconsideringtheHamiltonianH=S,whereisthestrengthofthespinorbitcoupling,Listheorbitalangularmoments,andSisthespinangularmomentum.Bydoingthis,wemustnowtakespinintoconsiderationasthespinandorbitalangularmomentaaremixed.ThiscausesarepulsionbetweentheP1+z;"andP1+x+iy;#Eenergylevels(aswellasothercombinationsthereof).ThismeansthattheP1+z;",P1+z;#,P2z;",andP2z;#energylevelsarepushedclosertotheFermilevel.Ifthespinorbitcouplingisstrongenough,theseenergylevelsendupcrossingtheFermilevel,asshowninFig.3.4(b)region(III).Thisresultsinabandinversionofthesetwoenergylevels,andsincetheP1+zandP2zstateshaveoppositeparity,theendresultisthetopologicalinsulatorphase.[17]ForBi2Se3andBi2Te3,thisbandinversionoccursonlyatthepoint,meaningthattheChernparityforthesematerialsis=1andthatasingularDiracconewillformattheexposedsurface.AnimportantdetailtonoteisthattheFermisurfacetakeninsidethebulkbandgapisnolongerdegenerateduetotheintrinsicspin-polarizationofthematerial.ThismeansthatenergiesaboveorbelowtheDiracpointwillharboroppositespinpolarizations,asshowninFig.3.5(a).ThisspinpolarizationalsoprotectstheTSSfromback-scatteringfromnon-magneticdefectsbecausenon-magneticdefectscannotspin,andthuscannotcausethequasiparticlestowintheoppositedirection.[19]TheexpectedtheoreticalbehaviorofthedensityofstatesforatopologicalinsulatorinsideofthebandgapisshowninFig.3.5(b)whenshownfromtheperspectivepresentedbyscanningtunnelingspectroscopy.Thisisthe29Figure3.4:(a)SchematicofthestructureforBi2Se3.Bi2Se3isalayeredmaterialconsistingofstackedquintuplelayersmadefrom3Seatomsand2Biatoms.ThelayersareattractedtooneanotherthroughvanderWaalsforcesandarethereforeasytoseparatebyexfoliation.(b)Schematicshowingthep-orbitalsplittingofBiandSeneartheFermilevel,whichisdenotedbythedottedblueline.Region(I)takesthechemicalbondingofBiandSeintoaccount,whileregion(II)furtherdemonstratestheontheenergylevelsfromthecrystalFinally,region(III)showstheofstrongspinorbitcoupling,inwhichbandinversionoccursandthetopologicalphaseiscreated.(a)isadaptedfromRomanowichet.al.and(b)isadaptedfromQiandZhang.[18,17]30Figure3.5:(a)DiagramoftheDiracconewithspinpolarizedconicalsectionsink-space.AtenergiesabovethecrossingpointoftheDiraccone,theelectronswillbepolarizedinonedirectionwhentravelingaroundthecone,whereasatenergiesbelowthepoint,electronswillbepolarizedintheoppositedirection.(b)BasiccalculationdemonstratingtheexpectedbehavioroftheDiracconewhenlookingatthedensityofstates.TocalculatetheexpectedDOScurve,weneedtointegrateovertheFermisurfaceintwo-dimensions.Theresultingintegralgivesalineardispersionrelationink,whichyieldsalinearDOSversusenergycurve.ThisiswhatwetheoreticallyexpecttoseewhentakingscanningtunnelingspectroscopyoverabareTI.primarysignatureoftopologicalsurfacestatesforthescopeofscanningtunnelingmicroscopymeasurements.3.2SuperconductivityIn1908,HeikeKamerlinghOnnessuccessfullyhelium,whichwouldallowforthecoolingofsamplesdownto4.2K.Threeyearsafterthisdiscovery,Onnesmadeanotherstrikingdiscovery.Whenmeasuringtheelectricalconductivityofmercury,hesawthattheresistanceofthemetalwenttozero.Thetemperatureatwhichthishappensisknownasthecriticaltemperature,TC,andisanintrinsicpropertyofthematerial.Formanyyears,31notheorywasdevelopedtoexplainthisverytneitherofthemacroscopicnormicroscopicscale.In1935,theLondonbrothersdevelopedaphenomenologicaltheorytohelpexplainthebehaviorofelectricandmagneticinasuperconductor.Moreimportantlyforthisdissertation,in1957Bardeen,Cooper,andSc(BCS)developedthecornerstonetheorytoexplainsuperconductivityonamicroscopiclevel,known,ofcourse,asBCStheory.[1]3.2.1BCSTheoryAkeyideaofBCStheoryisthatsomeelectronsinasuperconductorceasetobehaveassingleelectronsandformintopairsknownasCooperpairs.WhilesingleelectronsareforbiddenfromoccupyingthesamequantumlevelduetothePauliexclusionprinciple,Cooperpairsarenotbecausetheybehaveabosonsandcanthereforealloccupyasinglequantumstate.ItisimportanttounderstandnotonlythebehavioroftheseCooperpairsinourmaterials,butalsohowtheyform.LetusconsiderthemodelwheretwoelectronsareaddedtotheFermiseaatT=0K.Theelectronsareallowedtointeractonlywitheachother.Wecanthenwriteatwo-particlewavefunctionas 0(r1;r2)=Xkgkeikr1eikr2(3.3)wherethetwoelectronshaveequalandoppositemomentumkduetotheexpectationthatthelowestenergystateofthetwo-particlesystemwithhavezerototalmomentum.[20]Sinceweexpectanattractivepotentialbetweenthetwoelectronsandconsideringtheantisymmetry32ofthewavefunctionwithrespecttoparticleexchange,wecanrewritethewavefunctionas 0(r1r2)=hXk>kFgkcos(k(r1r2))i("##")(3.4)becausethesingletstatewillhavethelowestenergyandthecosinetermwillgivethehighestprobabilityfortheelectronstobenearoneanother.PluggingEq.3.4intotheSchroedingerequation,itcanbeshownthattheenergyeigenvaluesandweightingcotsgkcanbesolvedforwith(E2k)gk=Xk0>kFVkk0gk0(3.5)wherekaretheplanewaveenergiesandVkk0arethematrixelementsoftheinteractionpotentialbetweenthetwoelectrons.Inorderforsuperconductivitytoexist,theinteractionpotentialmustbenegativeorelsetherecanbenoboundstateinwhichtheelectronscanformaCooperpair.Thisnegativeinteractionpotentialstemsfromlatticevibrationsinthematerial.Roughlyspeaking,asoneelectronmovesthroughthelattice,itattractsmultiplepositiveions,whichinturnattractanotherelectron.Thesecondelectronpulledintothepositivepotentialpairswiththeinitialelectron,formingaCooperpair,asshowninFig.3.6.Inordertofullyunderstandwhatishappening,wemustinvestigatetheformationofCooperpairsandthethathasontheFermisea,sincewhentakingelectronicmeasurements,thatiswhatwearetrulymeasuring.BecausetheFermiseaiscomprisedofsomanyparticles,itisconvenienttoapproachtheformationofCooperpairsusingcreationandannihilationoperators.Inthisnomenclature,theoperatorcyk"wouldcreateaspinupelectronwithmomentum~kandck"wouldannihilate33Figure3.6:Astheelectronmovestotheright,itcausesadistortioninthelattice,creatingaregionwithhigherpositivecharge.Thisregionattractsasecondelectronintothenewlyformedpotentialwell,andthetwoelectronsbecomeapairknownasaCooperpair.thatsameelectron.Furthermore,aCooperpaircanthenberepresentedbythetermcyk"cyk#formedbyaspinupelectronwithmomentum~kandaspindownelectronwithmomentum~k.Usingthis,wecanproceedtotheBCSgroundstatewavefunction,whichisjGi=Yk(uk+vkcyk"cyk#)j 0i(3.6)wherej 0iisthevacuumstateoftheFermiseaandjukj2+jvkj2=1.Thelatterissuchthatjvkj2istheprobabilitythatthepair(k";k#)isoccupiedwhilejukj2=1jvkj2istheprobabilitythatitisnotoccupied.Havingdethisnomenclature,wecannowintroducethemostnotablecharacteristicofsuperconductorsinthisdissertation;thesuperconductingpairpotentialk.BCStheorydemonstratesthatthereisareductionofenergywithrespecttotheFermiseagroundstatecausedbythecondensationofCooperpairs.Inturn,thespectrumofsingle-electronstates,alsocalledquasiparticles,isgivenbyEk=q2k+˘2k(3.7)34where˘k=(~2k2=2m)EF.Here,thesuperconductingpairingpotentialkisask=Xk0Vkk0uk0vk0(3.8)whereVkk0isonceagainthematrixelementoftheinteractionpotential.Eq.3.8isknownastheselfconsistencyequationandemphasizesthefactthatthepairingpotentialstemsfromthequasi-particleelectronsandholes.[20]Finally,wecaneafewmorerelevantequationsforthisdissertation.ConsideringthatSTMmeasuresthequasi-particledensityofstates,itwouldbemostconvenienttoseetheexpectedformofthedensityofstatesintermsofk.Inthesimplestscenario,wecaninvestigatetheformforthedensityofstateswherethesuperconductorisnear0K,orT<<TC,isnotinamagneticandishomogeneousthroughout.kcanthenbeasapositive,realconstantwhichwewillasthebulkenergygap0.WecanthenwritetheBCSdensityofstatesasNBCSN0=8>>>><>>>>:EqE220whenE>00whenE<0(3.9)AtverylowtemperaturewecanexpectthedensityofstatestolooklikeFig.3.7.Thewidthofthesuperconductinggapis0andthesharpcuspsonbothsidesofthegapareknownascoherencepeaks,whicharefoundattheenergieswheretheCooperpairsbeginbreakingintoseparatequasi-particlesonceagain.Thecoherencelength,orlengthscaleofaCooper35pair,canbeintermsof0intheequation˘0=~vFˇ0(3.10)wherevFistheFermivelocityofthesuperconductingmaterial.WecanalsorelatethesuperconductingtransitiontemperatureTCtotheenergygapwiththeequations0=1:76kBTCintheweakcouplinglimit.[20,21]Figure3.7:AcalculatedsuperconductingenergygapdemonstratingwhattheexpecteddensityofstateswouldlooklikeonaBCSsuperconductor.Thewidthofthegapis0.Asmallamountofthermalbroadeningwasappliedinthecalculationtopreventthecoherencepeaksfromgoingtoy.363.2.2P-WaveSuperconductorsWhileSec.3.2.1givesasolidunderstandingofthecausesofsuperconductivity,itisnotquiteenoughtoexplainalloftheformsofsuperconductivity.Manysuperconductorsareknownas\s-wave"forwhichtheCooperpairsareassembledfromelectronswithanti-parallelspin,alsoknownasthespinsingletstate.ThespinstatesoftheCooperpairscanbeexpressedasSinglet=j"#ij#"iTriplet=8>>>>>>>><>>>>>>>>:j""ij"#i+j#"ij##i(3.11)Thes-wavesuperconductorsalsofeatureapairingpotential0thatisindependentofmomentumandissphericallysymmetric,muchlikethes-orbitalofanelectroninanatom.Asourunderstandingofsuperconductivitycontinuestogrow,wecontinuetomoreexoticsuperconductorsthathavepairingsymmetries.Themostrelevantformofexoticsuperconductivityforthisdissertationisp-wavesuperconductivity,whichispredictedtobeinducedinthetopologicalsurfacestateofatopologicalinsulatorwhenitisputincontactwithas-wavesuperconductor,duetothespinpolarizationofthetopologicalsurfacestateitself.Inordertofurtherdiscussp-wavesuperconductivity,letusdiscussthesphericalharmonicYmlthatrelatesdirectlytotheformofthepairingpotentialk).Forthespintripletsuperconductivity,weshouldinvestigatethesphericalharmonicsY11andY01.Condensingthesesphericalharmonicsdowntothex-yplane,asthetopologicalsurfacestateisitselftwo-dimensional,weseethatthepairingpotentialtakestheformofacloverleafin37plane,asshowninFig.3.8(a).Thisdemonstratesthatthepairingsymmetryinnotuniformforalldirectionsink-spaceandiszeroforcertainvaluesofk.Thiswillresultinthedensityofstatesnotbeingfullygappedlikethatofans-wavesuperconductor,asshowninFig.3.8(b).Themechanismbehindtheinducedp-wavepairingcanbeexplicitlyseenbylookingattheHamiltonianofatopologicalinsulatorwiths-wave(spinsinglet)superconductingpairing,denotedbytheequationH=H3DTI+Hs-wave(3.12)whereH3DTI=Zd2r y[iv(@x˙y@y˙x)] :(3.13)Equation3.13givesusthebandenergies(k)=vjkj,whichdescribestheupperandlowerbranchesoftheDiraccone.ByHs-wave=Rd2r " #+h:c:)describesthesingletsuperconductivitydiscussedinSec.3.2.1,andresultsinafullygappedquasipar-ticlesystemgivenbyE=p(k)2+2.ByperformingaunitarytransformationthatdiagonalizesthekineticenergyinH,wecanthenwritethisHamiltonianintermsofthecreationoperators, y,whichcreateelectronsinthepositiveandnegativeenergyregionsoftheDiraccone.ThisresultsintheexpressionH=Xs=Zd2k(2ˇ)2ns(k) ys(k) s(k)+h2kx+ikyjkj s(k) s(k)+h:c:io(3.14)wherewecanseethattheinducedsuperconductivityintheTSSproducesaformofpx+ipysuperconductivity.[22,23,24]Thiscanbequalitativelyunderstoodbythefactthattheinducedsuperconductivitypairselectronswithoppositemomentakandkbecausethey38carryoppositespinsasimposedbythespinorbitcouplingoftheTSS.TheCooperpairalsopicksupanangularmomentumbecausetheelectronspinrotatesby2ˇwhenencirclingtheDiraccone,andthuswehaveTR-invariantpx+ipysuperconductivity.[22]ThisspformofsuperconductivityispredictedtoharborMajoranafermions,whicharequasiparticlesthataretheirownantiparticles.Itcanbeshownthatbyapplyingamagneticnormaltothesurface,theCooperpairswillbeboundtoavortex,breakingTR-symmetryandallowingfortheexistenceofchiralMajoranaboundstates.[23,24,25]Figure3.8:(a)Aplotofthesuperconductingpairingpotentialk)forap-wavepairingsymmetryinthekxkyplane.(b)Thedensityofstatesofapurep-wavesuperconductor.TheDOSisnotfullygappedlikeinthes-wavecase,whichresultsina\v"shapeinsideofthesuperconductinggap.3.2.3SuperconductingProximityThetheoryfromsuperconductivitythatwemustunderstandforthisdissertationisthethatasuperconductorhasonanormalmetalwhenplacedindirectcontact.AsmentionedinSec.3.2.2,ans-wavesuperconductorplacedincontactwithaTIwillinduce39p-wavesuperconductivityinthetopologicalsurfacestate.Thegoalofthissectionisnottodiscussthecomplexphysicsbehindthatphenomenon,buttointroducethebasicmechanicsthatdriveinducedsuperconductivity.ThiscanbedonebyutilizingtheBogoliobov-deGennes(BdG)equations.RememberingthediscussionfromSec3.2.1,thesuperconductingpairingpotentialkarisesfromthequasiparticleexcitationsukandvkfromtheFermisea.Inthecaseofinducedsuperconductivity,wecantranslatethesefunctionsfrommomentumspacetopositionspaceduetothespacialdependenceonthesuperconductingpairingpotentialwhichwenowasr).Wecanalsoukandvkasu(r)andv(r),whicharethewavefunctionsforelectronsandholesrespectively.TheBdGequationsaretwocoupledSchroedingerequationswrittenasEu(r)=H0u(r)+(r)v(r)(3.15)Ev(r)=0v(r)+(r)u(r)whereEistheenergyeigenvalueandH0istheHamiltonianfortheFermiseaintheabsenceofthepairingpotentialwithrespecttoEF.[21,26]ReferencingonceagainbacktoSec.3.2.1,wehaveananalogtotheselfconsistencyequation3.8inpositionspacer)=g(r)Xnvn(r)un(r)(12f(En))g(r)F(r)(3.16)whereg(r)istheinteractionparameterandF(r)istheCooperpaircondensateamplitude.Fromhereitisdirectlyevidentthatr)isconstrainedbythesetwofunctions.Withthesuperconductor/normalmetalinterface,itisimportanttonotethebehaviorsofg(r)andF(r).BecauseF(r)isthemeasureofthelocalCooperpairdensity,itmustsmoothly40transitionfromthesuperconductingregionintothenormalmetalregion,ofcoursebecomingweakerastheCooperpairsmovefurtherfromthesuperconductor.Thissmoothtransitionresultsinaweakeningofsuperconductivityinthesuperconductorneartheinterface.F(r)containstwoimportantlengthscales;˘S,whichisthesuperconductingcoherencelengthmentionedinSec.3.2.1,and˘N,whichisthecoherencelengthoftheCooperpairsinthenormalmetal.Theinteractionparameterg(r)behavessomewhatdiscontinuouslyattheinterfacehowever.Thereisstillsomesmallscaleelectronscreeningattheinterface,sog(r)doesnotquitebehavelikeaperfectstepfunction,butforthepurposesofthisdiscussionwewilltreatitassuch.Fig.3.9demonstratesthebehaviorof(r)asafunctionofthepaircorrelationamplitudeF(r)andtheinteractionparameterg(r)acrosstheS/Ninterface.Inthesuperconductor,g(r)=1asthepairingpotentialisatit'smaximumof0.Inthenormalmetalhowever,g(r)<<1typically,allowingforinducedsuperconductivitytoexistinthenormalmetal.ItisimportanttonotethatthecalculatedforthesuperconductinggapspresentedinSec.4ofthisdissertationweredonebysolvingtheBdGequationstogetthedensityofstatesN(E)usingthemethodproposedbyGallagher.[27]41Figure3.9:Animagedemonstratingthebehaviorofr)asafunctionofthepaircorrelationamplitudeF(r)andtheinteractionparameterg(r).42Chapter4ProbingSuperconductor/TIInterfaces4.1SuperconductingProximityofPbBionBi2Se34.1.1MotivationTheinitialobjectiveofthisexperimentwastosolelymeasurethesuperconductingcoherencelengthinducedinBi2Se3byproximitytoans-wavesuperconductor.Thisspformofinducedsuperconductivityistheoreticallypredictedtoproducetwo-dimensional,TR-invariantp-wavesuperconductivityinthetopologicalsurfacestates.Thisformofp-wavesupercon-ductivitycanharborMajoranaboundstates,whichhavemanyinterestingcharacteristics.ApossiblegoalwastodetectaMaoranafermioninan\anti-dot"geometry,asshowninFig.4.1.Measuringthecoherencelengthrepresentsanimportantsteptowardsthisproposedexperiment.Asthemeasurementproceeded,wenotonlyextractedthesuperconductingcoherencelength,butalsoobservedtwounexpectedfeaturesintheLDOS:theoscillatorybehaviorintheLDOSoutsideofthesuperconductinggapneartheS/TIinterfacesandtheinverseproximityoftheTSSonthesuperconductingislands.ItwaswiththeseobservationsthatthefocusofthisworkchangedfromtryingtodetectMajoranaboundstatestotryingtofullyunderstandthecharacteristicsoftheseunexpectedbehaviorsinthesetopological43Figure4.1:Proposedanti-dotgeometrywhereacontinuoussuperconductingisdepositedonthesurfaceofaTIwithanti-dotsspacedevenlythroughoutthewhichwouldgoallthewaytotheTIsurface.ByapplyingamagneticeldnormaltotheS/TIsurface,Majoranaboundsstatesaretheoreticallypredictedtoformaroundtheboundaryoftheanti-dot,whichcouldbeprobedusingSTS.FigureadaptedfromtheHasanandKanecolloquiumontopologicalinsulators.[16]insulator/superconductorsystems.IncollaborationwithDr.LevchenkoandDr.Sedlmayr,wewereabletosuccessfullymodelanddescribethesesurprisingbehaviors.However,atthisstageweareunabletoconclusivelydecipherthespsuperconductingpairingsymmetryofthesystem.Toreiterate,thesimplesttheoriespredictthattheinducedsuperconductivityintheTSSshouldfollowp-wavesymmetry.Inreality,theinducedsuperconductivityislikelymuchmorecomplicated.Asthebulkofthetopologicalinsulatorisnotperfectlyinsulating(aswillbediscussedinSec.4.1.3),therewilllikelybeconductingstateswhichwillalsohaveinducedsuperconductingstatesthatdonotobeythepredictedp-wavebehavior.[18]Similarly,angular-resolvedphotoemissionspectroscopy(ARPES)measurementsdemonstratethepresenceoftrivialsurfacestatescausedbybandbendinginthesematerials.[28]SinceSTSisalocalmeasurementandissensitivetoallofthesetheLDOSmeasurementspresentedinthischapterarelikelyacombinationofboths-waveandp-waveformsofinduced44Figure4.2:LDOStaken40nmfromasuperconductingislandat4.2Kwiths-waveandp-waveDOScalculatedcurves.Atthistemperature,itistoseemuchbetweenthetwocalculatedcurvesasbroadeningcausethemtoappearverysimilar.FittingparametersareshownlaterinTable4.0a.superconductivity.Inanattempttoclarifywhatkindofinducedsuperconductivityweweremeasuring,wetheLDOSmeasurementswithboths-waveandp-waveDOScalculations.AnexampleoftheseattemptedisshowninFig.4.2.Ingeneral,wethatitiscultat4.2Ktotiatebetweenthes-waveandp-waveDOScurves,asshowninthenextsection;weconcludethatatpresentwecannotdiscernthesuperconductingpairingsymmetryinducedintotheBi2Se3.454.1.2IntroductionThree-dimensionaltopologicalinsulators(TI)[6],mostnotablyBi2Se3,Bi2Te3orBi1xSbx,wereonceknownprimarilyfortheirtendencytobegreatthermoelectricmaterials.Inrecentyears,amoreinterestingcharacteristicofthesematerialswasdiscovered;theirabilitytoharboratopologicallyprotectedelectronicsurfacestate.[29,30,31].Itisnowunderstoodthatthesetopologicalsurfacestates(TSS)stemfromacombinationofstrongspin-orbitcouplingandtimereversalsymmetryintrinsictothesematerials.Bystudyingtheinterplayofthesesymmetry-protectedstateswithsymmetry-breakinginterfaces,suchasmagneticorsuperconductingmaterials,wemaybeabletoobserveaplethoraofnewandprovidenewplatformsforpotentialtechnologicaladvances.Averydirectapproachtoachievethisgoalistostudyasetofphenomenaassociatedwiththesuperconductingproximityintheseexoticmaterials.Ithasbeenpredictedthatwhenatopologicalinsulatorisbroughtintocontactwithaconventionals-wavesuperconductor(SC),thesuperconductingproximityinducessuperconductingcorrelationsintotheTSSthathaveunconventionalp-wavesymmetry,discussedearlierin3.2.2[32,22].Thisoriginalresulttriggeredaodoffurthertheoreticalworks,somerepresentativeexamplesincludeRefs.[33,34,35,36,37,38],andamultitudeofexperimentalThelattercoversmanyvarioustechniques,spanningfromangle-resolvedphotoemissionspectroscopy,scanningtunnelingmicroscopy,pointcontact,andtialconductancemeasurements[39,40,41,42,43],toobservationsofsupercurrentsandunusualJosephsonFraunhoferpatterns[44,45,46,47,48,49,50,51,52];experimentsonphasecoherenttransportincludingmultipleAndreevFabry-PerotinterferometryandAharonov-Bohmoscillations[53,54,55,56].46Onthetechnicalside,proximitycanberealizedbygrowingthinlayersofSC-TIheterostructures,anapproachemployedbymostoftheexistingexperiments.OnecouldalsostudyinducedsuperconductivityintheTSSfromtheTI'sbulkwhichbecomessuperconductingwhenCuisintercalatedintoBi2Se3[57,58].WhileCuxBi2Se3retainstheDiracsurfacestate,itssuperconductingvolumefractionisrelativelylowwhichcausesobviouschallenges.HereweexploreacomplimentarybuttroutebyinducingsuperconductivityintoTSSlocallybydepositingamatrixofsuperconductingPbBiislandsonthesurfaceofBi2Se3.Thisapproachwastheorizedearlierasapathtocreatingsuperconductinggraphene[59].Itwasalsoemployedexperimentallytocreateatunablerealizationoftwo-dimensionalsuperconductivityinmesoscopicsuperconductor-normal-superconductor(SNS)arrays[60].Throughtheuseofcryogenicscanningtunnelingmicroscopy(STM),anexperimentaltechniquethatisidealforprobingthesuperconductingproximityininhomogeneoussamples,weareabletomeasuretheinducedlocalelectronicdensityofstatesinTIsurfacestates,asdiscussedin2.1.2.AschematicillustrationoftheexperimentisshowninFig.4.3(b),whereanSTMprobeisplacedneartheinterfaceofasuperconductingislandandaTI,andspectraaretakenwhilegraduallymovingtheprobeawayfromtheislandalongtheTIsurface.Inthissection,wepresenttheobservationoftwomainexperimentalresults.First,theSTMspectrarevealaclearsuperconductinggapinducedintotheTSSwhichdecaysaswemoveawayfromthesuperconductingisland.Fromthespatiallyresolvedprobesandofthegapfunction,weestimatethesuperconductingcoherencelengthtobeoftheorderof˘540nmalongthedirectionparalleltothequintuplelayers.Inaddition,atenergiesabovethegapweobserveoscillatorybehaviorofthedensityofstatesthatresemblestheTomaschinterference[63].Second,whilealltheexistingwereconcentratedonrevealingsignaturesofsuperconductivityinducedintotheTSS,muchlessattentionwaspaid47Figure4.3:(a)Aschematicofthestandardproximitypictureshowingtheinducedsuperconductingenergygap[61],ataninterfaceasafunctionofpositionx,whereFisthesuperconductingcondensateamplitude.Thepairinginteractionconstant,g,isgenerallytakentohavetheformofaclearstepfunction,butatsmallscales,thestepactuallyhasaslopeduetoelectronicscreening[62].(b)AschematicillustrationforthegeometryoftheSTMprobescanningoverthesurfaceofTIBi2Se3inproximitytoaPbBiisland.Inthisexperiment,weareinterestedinmeasuringthesuperconductingcoherencelengthintheplaneparalleltotheTIsurface,notinthenormaldirectiontothesurfaceasdoneinmanyotherexperiments.48tothecorrespondinginverseoftheTSSonanadjacentsuperconductor.ThisintriguingquestionisaddressesbytakingcarefulSTMdensityofstatesspectraonsuperconductingislandsanduncoveringtracesoftheDiracconethatseeminglyleaksfromtheTSSintothesuperconductor.Thisobservationmanifestlyprovidesevidenceforthepossibilityofaninversetopologicalproximity4.1.3ExperimentTheTIusedinthisexperimentisbismuthdopedBi2:04Se2:96,grownbyslowlycoolingastoichiometricmixtureofBiandSefromatemperatureof850C.FiveatomicplaneswithatomicorderSe1-Bi1-Se2-Bi1-Se1formaquintuplelayer(QL);theQLsareweaklyboundtoeachotherbythevanderWaalsforce,makingitpossibletoreadilyexposeapristinesurfaceforstudy.TheexposedQLsupportstheexistenceoftheTSS,whichfeaturesasingleDiraccone.WhileBi2Se3istypicallyntype,thebulkdopingofBitendstoshifttheFermilevelbacktothecenterofthebandgap[18].MorerecentmeasurementsintheTessmerlabdemonstratethattheshiftoftheDiracconemaystemmorefromthecleavingenvironmentofthesamplethanfromthedopinglevels.Pb0:3Bi0:7isusedasthesuperconductorduetoitsfavorablewhettingpropertiesonBi2Se3,itslargegapwidth(3.65meV),anditshightransitiontemperatureof8.2K[64].InitialmeasurementsusingpurePbasthesuperconductorareshowninAppendixA.ControlmeasurementsofthePb0:3Bi0:7areshowninFig.4.4,wherewemeasuretheexpectedsuperconductingDOSonathickofPb0:3Bi0:7andmeasureaveryreasonabletransitiontemperatureof8.3Kusingafour-probemeasurementconductedbyDr.RezaLoloee.FurthercharacterizationmeasurementsdonewithEDSweretakenonourPbBiwhichdemonstratedthatafterthermalevaporation,theresultingalloywas36%Pband64%49Figure4.4:(a)STStakenona250nmthickofPb0:3Bi0:7depositedonBi2Se3andwithacalculateds-waveDOSthathasbeenappropriatelybroadenedtoaccountforthermalandscattering(b)Four-probemeasurementtakenona400nmthickofPb0:3Bi0:7takenbyDr.RezaLoloee.Thetransitiontemperatureof8.3Kcloselymatchesthenominalvalueof8.2K.[64]Bothmeasurementstakenat4.2K.Bismuth,whichwhenaccountingfortheuncertaintiesofthemeasurementmethod(3%fromcontroldata)aswellastheevaporationtemperatureofPbandBi,demonstratesthatoursuperconductingalloyiswellwithintheexpectedratioatthesurface.AnexampleoftheEDSmeasurementstakenonourthickcontrolsampleisshowninFig.4.5.TheTIiscleavedinanitrogenenvironment,whereaTEMmaskisfastenedtotheTIsurface.Thesampleisthenplacedintoathermalevaporatorthatisalsopurgedinnitrogengas.Here,wedeposit10nmofPb0:3Bi0:7ontothesurfaceoftheTIbyevaporationthroughtheTEMmask.Thisresultsinlargearrayofsuperconductingislandswithadiameterof1.2m,asshowninFig.4.6.Fromthere,thesampleismovedtoourcryogenicBesockedesignSTMsystemformeasurement,wherethesystemispumpeddowntohighvacuumandcooledto4.2K.Itshouldbenotedthatallofthepreparatorystepsaredoneineitheranitrogenorvacuumenvironmentinordertominimizeairexposureandensurethatthesamplesurfaceremainsclean.AllSTMtopographsaretakenwithabiasvoltageof5Vandtunnelingcurrentof500pA,andallspectraaretakenwithavoltagedividerplacedonthe50Figure4.5:EDSmeasurementstakenonour250nmthickPbBishowingtheactualatomiccompositionofouralloy.EDSshowedthattheactualcompositionwas36%Pband64%Bi,whichiswellwithintheexpectedcompositionrangeforoursuperconductingislands.51biasvoltagelead(toimproveenergyresolution)overarangeof-45meVto45meVandmeasuredviaalock-inFigure4.6:Imagetakenunderopticalmicroscopeshowingthesuperconductingarraypatternafterthermalevaporation.ThissamplewasexposedtoairandwasthereforenotmeasuredunderSTM.Whendoingasurfaceprobemeasurement,itisimportanttofullycharacterizethesurfacebeingmeasured.Inthiscase,weareinterestedinthequalityofdepositionofthePbBiislandsaswellashowcleantheinterfaceisbetweentheislandedgeandtheTIsurface.InFig.4.7,weshowvariousatomicforcemicroscopy(AFM)andSTMtopographsdemonstratingthestructureoftheseislandsafterthermalevaporation.ThePbBidotsappeartobecomprisedofmany20-100nmradiusdropletsgrownontopofandaroundoneanother.ThiscanclearlybeseeninFig.4.7(a)and(b).Fig.4.7(c)isanSTMtopographshowingtheedgeofoneofsuchdropletformations,alongwiththerespectiveheightinFig.4.7(d).HerewecanseethattheinterfacebetweenthePbBiandtheTIsurfaceisveryabrupt,givingevidenceofminimalleakageofthePbBiontotheTIsurface.WeshouldalsonotethattheTIsurfaceisverysmooth,givingevidencethattherewasminimalairexposureorcontaminationofour52surface.Figure4.7:(a)and(b)Atomicforcemicroscopy(AFM)topographsofathermallydepositedPbBidot.Wecanclearlyseeaverygrainyappearancetotheoveralldot,butuponcloserinspectionitisclearthatthedotiscomprisedofmanysmallsuperconductingdroplets.(c)STMtopographofaPbBidropletwithitsrespectiveheighttraceshownin(d).TheradiusofcurvatureatthebaseofthedropletisanartifactduetotheradiusoftheSTMtip.TheactualinterfacebetweenthedropletandtheTIisverysharp,whichisidealforourmeasurement.Scalebars:(a)500nm(b)100nm(c)40nm.Wepresentthelocaldensityofstates(LDOS)measurementstakenviacryogenicSTMatatemperatureof4.2KinFig.4.8.TheleftpanelshowsaseriesoftialconductanceplotstheLDOSoftheTSStakenatvariousdistancesawayfromthesuperconductingisland.Atadistanceofapproximately40nm,theinducedsuperconductinggapisroughly20%smallerthanthecorrespondinggaponanisland,whileatdistanceoforder200nmthe53gapfallstoalmosthalfofitsvalue.Atadistanceof>5m,theDiracconeistheonlydominatingfeatureintheLDOSsincethelocalregionoftheTIisnolongerwithintherangeofthesuperconductingproximityt.TheDiracconeisnotperfectlysharpasthereisstillasmallamountofthermalenergyat4.2K,whichbroadensthepointofthecone.Again,itshouldbenotedthattheDiracconeis\raised"andcannotgoto0duetotheeverpresentcontributionofthebulkstatesfromtheTI.[18]AsmentionedinSec.4.1.1,theinducedsuperconductivityislikelyacombinationofs-waveandthepredictedp-wavepairingsymmetries,butat4.2Kweareunabletotiatebetweenthetwo.Fig.4.9showstheinducedsuperconductinggapwidthasafunctionofdistancefromthePbBidroplet.Wethedatawithanexponentialdecaytoextractacoherencelengthof˘540200nm.ThegapwidthisextractedfromtherawdatabytakinglinearextrapolationsofthedI=dVcurvesaroundofthesuperconductinggapandsubtractingtheextrapolatedlinesfromthefulldata.ThisenhancesthecoherencepeaksonbothsidesoftheFermilevelandallowsustoextractthevalueofAsacheck,wealsoapplieds-waveBCScurve(seeFig.3.7)andapproximatep-wavecurve(seeFig.3.8(b))tothedata;thebestsgavereasonableagreementwiththelinearextrapolationmethod.Theparametersforthes-waveandp-wavecalculationsareshowninTable4.0abelow.Thep-waveDOSrequiredlessbroadeningthanthes-waveandassuch,weinterpretedthesetobebetterthantheconventionals-waveWecomparethecalculatedandextractedvaluesofinTable4.0bbelow.Thatbeingsaid,thesecalculationsdonotconclusivelydemonstratethesppairingsymmetryinducedinthesurfacestatesofthetopologicalinsulator;rather,thisservestodemonstratehowitistoextractthesppairingsymmetriesfromLDOSmeasurementsat4.2K.Bycoolingthesampletolowertemperatures(suchas300mK),itmaybepossibletogainabetterunderstandingofthese54OnIsland(s-wave)40nm(p-wave)40nm(s-wave)160nm(p-wave)160nm(s-wave)(meV)4.03.02.11.91.3(meV)1.300.751.601.401.42L(nm)795.321,378.56646.19738.51728.11Table4.0a:TablepresentingthecalculatedvaluesofandotherparametersforthebestDOScurvesatvariousdistancesfromthesuperconductingisland.Thealgorithmutilizedboththermalandenergeticsmearing.HereisthecharacteristicenergysmearingofthematerialandListherelatedmeanfreepath,andthetwovaluesarerelatedbytheequationL=hvf=Allcalculationsweredoneusingatemperatureof4.2K.Thep-wavecalculationsweredoneviaasumofseparates-wavesuperconductingcurvesweighedaccordinglybyap-waveorbitalfunction.BestFit(meV)Extracted(meV)(meV)OnIsland4.04.00.0040nm3.03.250.25160nm1.92.550.65Table4.0b:AcomparisonofthegapwidthstakenfromthebestDOScurveswiththewidthsextractedusingourlinear-subtractionmethod.WealsoobservedoscillatorybehaviorintheLDOSneartheS/TIinterface.Onthedatasettakenat40nmawayfromtheislandoneseesoscillatoryfeaturesoccurringwiththeperiodofroughly5mV.Interestingly,similarfeaturesarepresentontheLDOSplotstakenonthetislands,whichareshownontherightpanelofFig.4.8.WhilepurePbisknownforhavingphononmodesoutsideofthesuperconductinggap,itshouldbenotedthatthespPbBialloythatweusedoesnotexhibitthissamebehavior.[64]Furthermore,inadditiontooscillationsatenergiesabovethegap,tracesoftheDiracconearealsorevealed.WeattributethisphenomenontotheinversetopologicalproximitywhereLDOSpropertiesoftopologicalsurfacestatespenetrateintothesuperconductingisland.Inwhatfollows,thisbehaviorisillustratedusinganeleganttheoreticalmodeldevelopedbyDr.AlexLevchenkoandDr.NicholasSedlmayr.55Figure4.8:TheleftpanelrepresentsdI=dVcurvesmeasuredat4.2KtakenatvariousdistancesfromaPbBiisland.TheLDOSdisplaysclearsignatureoftheinducedsupercon-ductinggap,withthearrowsdenotingthelocationofthecoherencepeaks.TheonlyfeatureatdistancesfarfromthesuperconductingislandistheDiraccone.Anothernotablefeatureofthepresenteddataisvisibleoscillationsoutsideofthesuperconductinggap.TherightpanelrepresentsdI=dVcurvesmeasuredontislandsatnominallythesameconditions.ThedatashownheredemonstratesanincreasingDOSforbothpositiveandnegativevoltagesoutsideofthegap,asopposedtotheDOSpredictedbyBCS.ThisisindicativeofDiracconestatesinthesuperconductor.AlldataarenormalizedsothatdI=dVj20mV=1.56Figure4.9:TotalenergygapwidthmeasuredbySTMat4.2KwiththeexpectedexponentialdecaydescribedbythefunctionGapWidth=0ex=˘,demonstratingthatastheprobemovesfurtherfromthePbBidroplet,thesuperconductinggapdecreasesinwidth.Herethebesttgivesapairingpotentialof0=3:640:40meVandacoherencelengthof˘=540200nm.Inordertoextractforeachdatapoint,wetooklinearfrom-6{-1mVand1{6mVofthedI=dVcurveandsubtractedthemfromthefulldatainordertoenhancethecoherencepeaksonbothsidesoftheFermilevel.Theinsetshowsthedataandcurveoveranexpandeddistancerange.574.1.4TheoryandCalculationsIthasbeenemphasizedinpreviouschaptersthatcrystalsofBi2Se3possesstopologicalsurfacestates,aswellasintra-gaptrivialconductingstatesinthebulkoriginatingfromunintentionaldoping,andimportantlyalsoatthesurfaceduetothebandbending[28].ThiscoexistenceoftopologicalandtrivialsurfacestatesleadstocertaincomplicationsinthecontextoftheproximityInparticular,phasecoherenttransportmeasurements[47,55,56]suggestthatthesuperconductingproximityoftrivialstatesisinthedominatedtransportregime,whereastopologicalstatesdisplaytransporttsthataresptotheballisticdomain.Inordertogainsometheoreticalinsightintothesystem,theexperimentalset-upismodeledintwolimits.Asaapproachwehaveconsideredtheelimitthatshouldberelevanttoapartoftheproximitygovernedbythetrivialsurfacestatesstemmingfromtheconductingbulkaswellasthetrivialsurfacestatefrombandbending.Incompleteanalogytopreviouslystudiedmesoscopicsuperconductor-normalproximityjunctions[65,66]wesolvedthestandardUsadelequationforacirculargeometrydescribingasuperconductingislandofradiusRsurroundedbyannormalsystem.WithinthisformalismtheproximityisdescribedbythesemiclassicalGreen'sfunctionG(x;!)=cos[(x;!)]ofpositionandenergythatinthenormalregionobeysthenonlinearequation@2@x2+1x@@x+i!EThsin=0(4.1)wherex=r=RisthedimensionlesspositioncoordinateandETh=~vFl=2R2istheThoulessenergybytheFermimomentumvFandmeanfreepathl.TheLDOSisobtained58fromtherealpartoftheGreen'sfunction(x;!)=0Recos[(x;!)];(4.2)where0isthenormaldensityofstateswithouttheproximityInalinearizedregime,applicableatdistancesawayfromtheboundarywheretheproximityisweak,ananalyticalresultforEq.(4.1)ispossible,(x;!)=0(!)K0(xpi!=ETh)K0(pi!=ETh);(4.3)elsewheretheproblemmustbesolvednumerically.InEq.(4.3)K0(z)isthemoBesselfunctionand0=cos1(BCS0)andBCSistheBCSdensityofstatesontheisland.Thisanalysispredictsareasonables-wavelikeproximityinducedgapEginthenormalregion,seeFig.4.10(a),albeitwithascalesetbytheThoulessenergy,Eg˘ETh,ratherthanasuperconductinggapHoweveritcontainsnospecinformationaboutthemicroscopicsurfacestatestructureofthematerial,whichisneglectedinthesemiclassicaldisorderedlimit.ForthetypicalknownvaluesofvF'5105m/sandl'80nmwhicharesptoBi2Se3surfacestates,andR'500nm,oneestimatesaproximityinducedThoulessgapEg.1meVtobeinaproperparameterregimewhencomparedwith67]Basedonthismodelingtheexpectedsuperconductingcoherencelengthfordisorderedsurfacestatesisintherangeof˘=pvFl=˘200nm.However,inordertotheactualspatialofthedecayofthegapfunctionx)intheBi2Se3foundintheexperiment[seeinsetinFig.4.10(b)]itisnecessarytouseaparameteroftheThoulessscalewhichistfromwhatisestimatedabove.Theoscillatoryfeaturescanalsonotbereproducedinthis59limitas,althoughinprincipletheresultingLDOSinEq.(4.2)isoscillating,itfollowsfromEq.(4.3)thattheoscillationanddecayscaleoftheBesselfunctionarecontrolledbythesameparameter.Inordertodescribethetopologicallynontrivialsurfacestatesandtheoscillatorybehaviorobservedinourmeasurements,wethenconsideredtheballisticlimitoftheproximityThiswasdonebysolvingtheGor'kovequationsinaplanargeometrywithatwo-dimensionalplaneconsistingoftopologicalsurfacestateswiths-wavepairingimposedononehalfoftheplane,forx<0.TheresultsofthesecalculationscanbedescribedbyBesselfunctions,whichgiveaverygooddescriptionoftheoscillationsseenintheexperimentalresults.Inthesuperconductingsideofthismodel,thetheoreticalresultsdemonstrateaLDOSwithanoscillatory,Tomasch-likenature.Onthenormalsideofthemodel,thecalculationsdemonstrateaLDOSwithoscillatory,Friedel-likenature.Eventhoughthismodelingwasdoneforatgeometrythenthatofoursystem,webelievethattheseoscillationsaregeneric.Additionalanalysisshowsthatforthesphericalsymmetry,theGor'kovequationsleadtoaLDOSofamorecomplicatedform,resultinginBesselfunctionswiththarmonics.Regardless,theessenceofthetremainsthesame.4.1.5ConclusionsBydepositingsuperconductingislandsonthesurfaceofatopologicalinsulator,wewereabletomeasureinducedsuperconductivityinthesurfacestatesoftheTIthroughthesuperconductingproximitythroughtheuseofcryogenicSTM.Wemeasuredthesizeofthesuperconductinggapasafunctionofdistancefromthesuperconductingisland,andcouldnotonlyseethesuperconductinggapshrinkaswemovedfurtherfromtheisland,butwerealsoabletoobtainanestimatedcoherencelengthontheorderof500nm.We60Figure4.10:(a,b)Thelocaldensityofstatesintheelimitasafunctionofenergy(a)attdistancesfromtheboundaryof0:5R;1R;1:5R,shownbythered,black,andbluecurvesrespectively,andasafunctionofposition(b)atenergies0:1ETh;0:8ETh;1:5ETh,onceagainshownbythered,black,andbluecurvesrespectively,withlinesrepresentingthelinearizedanalyticalresult,andsymbolsthenumericalresultoffullnonlinearEq.(4.1).Theinsetshowacomparisonbetweenthegapfoundexperimentally(bluediamonds)andtheanalyticalresult(solidline).(c)ThelocaldensityofstatesasafunctionofenergyforthetopologicalinsulatorsurfacestatestotheleftoftheTI/Sboundaryat0,whichistreatedasbulksuperconductor.(d)ThelocaldensityofstatesasafunctionofenergyforthetopologicalinsulatorsurfacestatestotherightoftheTI/Sboundaryat0,whichistreatedasbaretopologicalinsulator.Aphenomenologicaldampinghasbeenincluded,andthebulkdensityofstatesareincludedasacomparison.61werealsoabletoobserveoscillatorybehaviornearthesuperconductor/TIinterface,whichtheoreticalmodelingpredictsiscausedbyFriedel/Tomasch-likeoscillations.Furthermore,wesawevidenceoftheDiracconeinducedonthesuperconductingislands,indicatingthatthereexistsaninverseproximityinwhichthesuperconductorisalsobythetopologicalsurfacestate.4.2SuperconductingProximityofPureNbonBi2Se34.2.1IntroductionThissectionpresentsfurtherevidenceofthesuperconductingproximitywhichfollowedfromacollaborationwiththeVanHarlingengroupattheUniversityofIllinoisinUrbana-Champaign,IL.ThemainobjectiveofthiscollaborationistoutilizetheirmasteryofS-TI-SJosephsonjunctionfabricationinordertoimagethesuperconductingTSSinsideofaJosephsonjunctionusingbothSTMandSCAI.AgreatdealofhasgoneintofabricatingtheidealsamplesforimagingunderSTM,andthesamplefabricationmethodutilizestheirabilitytomilltrenchesdeepintotheTIsubstrateandthermallyevaporateNbinpatternswithathicknesslargerthanthecoherencelengthforNb.ThisensuresthattheNbwillbefullysuperconductingandthatwewillgetthebestresultsinthesemeasurements.Fortheinitialstudy,wedecidedtotakemeasurementsaroundtheedgesofthearrayswherethetipcouldbepositionedfarfromtheNb.ThisallowsustoreproducetheexperimentwithPbBiislandsnowusingNbasthesuperconductor.ThegoalistomeasuretheinducedsuperconductivityintheBi2Se3inordertogainfurtherknowledgeoftheinverseproximity62Figure4.11:(a)AschematicillustrationforthegeometryoftheSTMprobescanningoverthesurfaceofTIBi2Se3inproximitytoaNbisland.(b)Aschematicofthestandardproximitypictureshowingtheinducedsuperconductingenergygap[61],ataninterfaceasafunctionofpositionx,whereFisthesuperconductingcondensateamplitude.Thepairinginteractionconstant,g,isgenerallytakentohavetheformofaclearstepfunction,butatsmallscales,thestepactuallyhasaslopeduetoelectronicscreening[62].andoscillationsseeninSec.4.1.AschematicillustratingtheexperimentisshowninFig.4.11,withalargescaleSEMscanofthesampleshowninFig.4.12andanAFMtopographofoneofthesuperconductingarraysshowninFig.4.13.AnimportantrencetonotebetweenthesetwoexperimentsisthatinSec.4.1,weusedacleavedBi2Se3crystal,whereasforthisexperiment,theBi2Se3substrateisepitaxiallygrownonsapphire.Inthissection,wewillshowtheresultswewereabletoobtainwhenthesamplewascooledbelowthesuperconductingtransitiontemperature.4.2.2ResultsItisveryimportantthatweestablishthatthereisahighqualityDiracconeintheBi2Se3beforemeasuringthesuperconductingproximityWedidthisbywalkingthesamplefarfromtheNbarraysandtakingspectraoveracleanBi2Se3region.InFig.4.14we63Figure4.12:SEMimagetakenbycollaboratorCanZhangshowingaregionofNbarrays.Inthissampleweareprimarilygivensquareandhexagonalarrayswithddistanceinbetweenislands.Thesamplealsocontainsanti-squarearrays,circulararrays(likethoseusedpreviouslyinourPbBiandPbmeasurements),tri-junctions,andloopsjunctions.TheloopjunctionswouldallowustothreadamagneticthroughasuperconductingloopandmeasuretheresultfromtheinducedphaseintheNbjunction.ThearraysarenecessaryinordertoeliminatetheneedleinthehaystackprobleminherentinSTMmeasurements.64Figure4.13:AFMtopographtakenbycollaboratorCanZhangshowinganarrayofhexagonalNbislands.ThisdemonstratessmoothNbislandswithacleanTIsubstrate.canseeaclean,sharpDiracconethatcomesclosetozeroaroundtheFermilevel,whichsohappenstobeatapproximately0mV.ThisisindicativethattheTIisbehavingaspredictedfarfromtheNbarraysandthatanyinducedsuperconductivityshouldbehappeningwithinthebandgapoftheTI,andthereforeinthetopologicalsurfacestates(aswellasinthetrivialsurfacestates).Wewerealsoabletoacquireafewproximitymeasurementsasafunctionofdistancefromthesuperconductingarrays.InFig.4.15weshowameasurementshowingthegapdecreaseinsizeaswemovefurtherfromthearray,butthenbeginningtoincreaseagainasthetipgetsclosertoanotherarray.Aswecanoftentimesseeinregionsnearsuperconductingislandswithinducedsuperconductivity,wealsosawmultipleinstancesofveryclearTomasch/Friedel-likeoscillations.InFig.4.16weshow2exampleswhere65Figure4.14:MeasurementoftheDiracconetakenfarfromtheNbarrays.Thisspectrumwastakenat1.78K.Webelievethattheconeisnotperfectlysharpduetothesmallamountofthermalbroadeningatthistemperature.66oscillationsareveryprevalentoutsideofthesuperconductinggap.Unfortunately,duetoproblemswithtipstability,wewereunabletoidentifythedistancefromthesuperconductorforthesespectra.Interestingly,thetopcurveofFig.4.16looksverynearlylikeaDiracconewiththeoscillationsimposedonthelinearDOS.ThiscouldstemfrombeingnearaNbislandwithweaksuperconductivity,assuperconductivitymustbepresentinorderforthesetobeobserved.Finally,andpossiblythemostinterestingdatathusfarfromthesesamples,weobservedstrongerevidenceoftheinverseproximityontheNbislands.Thatistosay,weseeclearevidenceoftheDiracconewhentakingspectraontheNbaswellasonthebareTIsurface.ThisdemonstratesthatthisphenomenonisnotjustseeninPbBisuperconductingislandsandthatthemeasurementspresentedinSec.4.1likelydonotstemfromtheintrinsicspin-orbitcouplingfoundinbismuth.[67]Fig.4.17showseseparatemeasurementsofthisphenomenonwheretheDiracconecanbeseenoutsideofthesuperconductinggap.Whatisalsosurprisingisthelackoftheclearcoherencepeaksexpectedtoexistinapures-wavesuperconductoratsuchlowtemperatures(<2K).Wehaveafewofthecurveswithcalculateds-wavegapsthathavebeenthermallybroadenedandwithscattering.SincesomeofthesuperconductinggapsaremuchweakerthanotherstakenonNb,itispossiblethatBCSaloneisnottfordescribingthisphenomenon.Furthermore,wehavealsotakenmeasurementsatroomtemperature.AsshowninFig.4.18,theTSSappearstoleakintothenownormalNbislands,indicatingthattheinverseproximitydoesnotrequiresuperconductivityinorderforittooccur.Dr.NicholasSedlmayrhascompletedsomeverycompellingcalculationsthatdemonstrateourinterpretationofthisdata.WebelievethatthereissomehybridizationoftheTSSwiththenormalmetallicstatesoftheNb,whichisfurtherreinforcedbythecalculatedLDOS67Figure4.15:Proximitymeasurementtakennear1mofaNbarray;thedistancesfromtheedgeofaNbislandareindicatedintheplot,wheretheredarrowsnotethelocationofthecoherencepeaks.Wecanclearlyseethegapdecreaseinwidthaswemoveawayfromtheisland,thenbegintoincreaseagainaswearemostlikelygettingclosertoanotherarray.Measurementtakenat1.6K.68Figure4.16:TwospectratakenattpointsdemonstratingTomasch/Friedel-likeoscillationsoutsideofthesuperconductinggap.Interestingly,thetopcurvelooksliketheoscillationsaresimplysuperposedonanormalDiraccone,butsincesuperconductivitymustbepresentinorderforthetobeobserved,itispossiblethatthismeasurementwastakennearaweakNbisland.Measurementsweretakenat1.6K.69Figure4.17:VariousspectratakenontNbislandsdemonstratingtheinverseproximityoftheTSSonthesuperconductor.Intwocurves,weappliedusingcalculateds-waveDOScurvesthatbeenthermallybroadenedandhadscatteringapplied,wherethevaluesoftheseparametersareshownintheplot.Thecalculatedcurveshadatemperatureof1.6K,acoherencelengthof38nm,andathicknessof60nm.Surprisingly,wecannotseeclearcoherencepeaksinthesuperconductinggapspotentiallyduetotheinducedTSSformingp-wavesuperconductivityintheNbislands.Allmeasurementsweretakenattemperaturesbetween1.5-1.8K,wherethesuperconductinggapisexpectedtobebothverydeepandtohavesharpcoherencepeaks.ThetopcurveisaDiracconemeasuredinaregionofthesamplewherenoNbislandshadbeendeposited.70Figure4.18:Room-temperaturespectratakenonandNbislands.The3curvesontheleftweretakenonNbislandsfoundinthesamearraywhilethecurvesontherightweretakenat3separatelocationsonbareBi2Se3.WebelievethatthisisthesignatureoftheDiracconeonNbatroomtemperature,whichindicatesthatthisinverseproximityislikelynotdependentonthesuperconductingstateoftheNbandthereforemuststemfromsomeothermechanism.71curvesshowninFig.4.19.OurmodeldemonstratesthethereisanintricateinterplayoftheTSS,thenormalmetallicstates,andBCSpairing.Outsideofthesuperconductinggap(andofcourseaboveTC),theNbstatesbehaveasanormalmetalandallowfortheTSSto\leak"intotheNb.ThecalculationspresentedbelowcomefromanexactanalyticalsolutionofthesystemHamiltonianandanumericalsolutionfortheenergydispersion.ThesystemHamiltoniancanbedescribedbyH=HM+H+HTSS+Hc(4.4)whereHMistheHamiltonianforasimple,clean2Dmetal,Hrepresentsthes-wavepairingofthosemetallicstates,HTSSrepresentsthetopologicalsurfacestateswhichhaveleakedintothesuperconductingislands,andHcisalocalcouplingtermbetweenthetopologicalsurfacestatesandthenativemetallicstates.SpfortheNb-Bi2Se3system,HM;=Rd2ryrHM;r,whereristhespacialcoordinate(x;y)intwodimensions,HM+H=^˘˝z+˝xintheNambubasis,yi=fcyr";cyr#;cr";cr#gandacorrespondingwave-function TrwrittenintermsoftheBogoliubov-coherencefactors:fur";ur#;vr";vr#g.Theoperatorcyr˙createsaparticleofspin˙atpositionr.Here,weuse~˝asthePaulimatricesactingintheparticle-holesubspaceand~˙asthePaulimatricesinthespinsubspace.Thequasi-particledispersionisassumedtobequadraticforlowmoment^˘=2=2mwhereisthechemicalpotential.HM;iscompletelydiagonalinspin.TheHamiltoniandescribingthetopologicalsurfacestatescanbedescribedbyHTSS=Rd2r˜yrHTSS˜r,whereHTSS=(ivFr~˙TSS)˝zisactingon˜yr=fayr";ayr#;ar";ar#g.Theoperatorayr˙createsaparticlewithspin˙atpositionr.Finally,thecouplingHamiltonianHccanbedescribedbyHc=Rdrr[˜yr˝zr+h:c:],whichrepresentsalocalspin-independent72hybridizationdescribedbytheparameter.Thenumericalsolutiontotheenergydispersionisdescribedbytheequationk=p2r22+2+˘2k+2k+q2+˘2k2k]2+422+(˘k+k)2](4.5)wherephysicallyaccountsfortheparticle-holesymmetryoftheenergy,accountsforthetspinbands,accountsforthemixingoftheTI/Selectronspecies,isthesuperconductingpairingpotential,isthetunablepairingstrengthbetweentheTSSandnormalmetalstates,˘kisthedispersionofthemetallicstates,andk=kvFTSS,wherevFistheFermivelocityfortheTSSandTSSisthechemicalpotentialfortheTSS.Mathematically,;;=1toaccountforall8possibleenergybands.When!0,thedispersionrelationrecoverstheBCSandTISSdispersions.TheDOScanbecalculatedfromthisenergydispersionbynumericallysolvingtheequation()=Z10dkk2ˇe(k)22pˇ(4.6)whereisabroadeningenergy.Interestinglyenough,thebestdatainthesecalculationsoccurswhensuperconductivityisinducedinboththe\native"electronstatesaswellastheleakingTSSinthesuperconductorviathecouplingterm.AsshowninFig.4.20,verygoodusingthemodelcanbegenerated.Fig.4.20(a-c)allutilizevF,,andabroadeningtermasfreeparameters,whileFig.4.20(d-f)utilizevF,,andasfreeparameters.BycarefullytuningweacquirebetterastheproximityofthesuperconductortotheTIaswellastheleakingsurfacestateswillofcoursethevalueofinthesuperconductingmaterial.Animportant73Figure4.19:LDOSdatawiththeoreticalcurvesbasedonamodelthatincludesnativemetallicstatesandproximitycoupledTIsurfacestates.Model1(blue)hasnoBCSpairing.Model2(purple)hasBCSpairinginthenativemetallicstatesonly.Model3(red)haspairinginboththenativeandtheproximity-coupledTIstates.tonoteisthattheseutilizeavFontheorderof˘100m/s.ThisisgreatlyreducedfromtheexpectedvFofBi2Se3,whichisvF=5105m/s.[31]WebelievethiscanbeattributedtothefactthattheTSSinthesuperconductorisspreadingperpendiculartothedirectionofelectronw",thusdramaticallydecreasingtheFermivelocityvF.Thiscanbethoughofasthequantumanalogofaexpandingfromasmallreservoirintoalargerreservoir.4.2.3ConclusionThisexperimenthasbeensuccessfulinthatwewereabletomeasureinducedsuperconductivityintheBi2Se3TSSaswellasaninducedDiracconeintheNbsuperconductor.NotonlydowemoreconvincingevidenceoftheinverseproximityofaTIonasuperconductor,wealsoassembledamodelwhichcannicelydescribethisphenomenon.Bytuningvariousparametersinthetheoreticalcalculation,weseeveryaccuratetothedata,whichhelpto74Figure4.20:ExperimentaldataoftheDOStakenon3separateislandswithcalculateddemonstratingtheleakageoftheTSSintothesuperconductingNbislands.(a-c)allutilizevF,,andabroadeningtermasfreeparameterswhile(d-f)allowtobeafreeparameterinthecalculationaswell.Ingeneral,(d-f)havebetterasitisexpectedthatwillbebyboththeproximitytotheTImaterialaswellastheTSSstatesleakingintothesuperconductor.75verifyourcurrentinterpretationofthisinverseproximity76Chapter5Conclusions5.1SummaryofResultsTheprimaryscopeofthisworkwastoutilizescanningtunnelingmicroscopyandsubsur-facechargeaccumulationimagingtomeasurethevariousproximityinducedthatsuperconductorsandconventionalinsulatorshaveontopologicalinsulators.Intermsofsu-perconducting/topologicalinsulatorinterfaces,wesuccessfullymeasuredthesuperconductingcoherencelengthaswellasobservedmultipleunexpectedbehaviors,bothinthedensityofstatesofthetopologicalinsulatoraswellasthesuperconductor.Weobservedoscillatorybehaviorneartheinterfaceofthesetwomaterials,whichwebelievetobeFriedel/Tomasch-likestatesenhancebyAndreevattheboundary.Wealsoobservedtherathersurprisinginverseproximityofatopologicalinsulatoronasuperconductor,bothinthenormalandsuperconductingphases.ThefactthattheDiracconecanleakintothenormalconductingstatesofasuperconductorcanleadtoamultitudeofinterestingandunexpectedresultsinthefuture.5.2FutureGoalsOurTI-superconductorproximityemeasurementsshowclearevidenceoftheDirac-cone-likestatesthatappearinthesuperconductor,whichwecalltheinverseproximityAn77importantquestionraisedbytheseobservationsisthedegreetowhichthesestatesretaintheirtopologicalcharacter.Inotherwords,isthereane\topologicalcoherencelengthonthesuperconductingsideoftheinterfacethatsetsthedistanceoverwhichmomentumandspinwillbelocked?Toexplorethisquestion,inthenearfutureweplanaseriesofSTMmeasurementsoncontinuousNbofvaryingthicknessesgrownonBi2Se3.WewillcharacterizethedegreetowhichDiracconestatesappearintheDOSandresolvespatialpatternsinthedefectscattering,similartotheSTMmeasurementsperformedbytheYazdanigrouponTIsurfaces.[68]TheTessmergroupisalsointheprocessofmovingtowardsanotherexcitingexperimentwhichwillprobethelandscapeofthecriticalcurrentinS/TI/Sjunctions.ThisprojectwillcomefromcontinuedcollaborationwiththeVanHarlingengroupandwillutilizeallofthemethodspresentedinthisdissertation.AschematicoftheproposedexperimentisshowninFig.5.1.WewillusetheSCAImeasurementmethodandthetopologicalinsulatorproximitydiscussedinAppendixBintheseS/TI/SjunctionstotunethelocationofthesupercurrentandtoobservethepathoftheJosephsoncurrentinthex-yplane,aswellastheverticalpositionsofthechargecarriers.78Figure5.1:SchematicsoftheproposedmeasurementwhichwillutilizeSCAImeasurementstomeasurethepathoftheJosephsoncurrentinabiasedS/TI/Sjunction.79APPENDICES80AppendixASuperconductingProximityofPbonBi2Se3IntroductionPriortoourexperimentsutilizingaPbBialloyforoursuperconductingmaterial,wedidextensivemeasurementsusingpurePbasthesuperconductoraswell.Whilethisworkdidnotdirectlymakeitintoapublication,itisimportanttonoteourresultsfromthesedatasetsaswellandtounderstandwhywemovedtothealloyedsuperconductorPbBi.ItisduringthesemeasurementsthatwewereabletosolidifyoursamplepreparationmethodsaswellasestablishsomeoftheimportantpreliminaryresultsandmeasurementtechniquesusedtogetthedatapresentedinSec.4.1.ExperimentInourpreliminaryexperiment,westillutilizedBi2:04Se2:96astheTIsubstrate.ThepreparationmethodsareidenticaltothosefromSec.4.1withtheimportantnotethatweusedPbinsteadofaPbBialloyforthesuperconductor.Weinitiallythoughtthistobeanidealmaterialasitalsohasareasonablyhightransitiontemperatureof7.25Kandanenergygapof2.61mV.[64].Acontrolmeasurementofa400nmthickPblayerdepositedonthe81FigureA.1:STStakenona400nmthicklayerofPbdepositedonthesurfaceofBi2Se3.Herewecanseeaveryclearsuperconductinggapaswellassignaturesofphononoscillationmodesoutsideofthesuperconductinggap.Thismeasurementwastakenat4.2K.surfaceofBi2Se3isshowninFig.A.1,wherewecanseeaveryclearsuperconductinggapwithawidthofaround2.8mV,whichwhenaccountingforbroadeningcts,isaroundtheexpectedvalueof2.61mV.Ofcourse,wealsoconductedsomebaselinemeasurementsintermsofPbdepositionquality.Fig.A.2showsanAFMtopographofaPbislandarray.Wecanonceagainseethat,likethePbBiislands,thePbislandsalsocompriseofsmallerdroplets.WhendoingmeasurementsintheSTMsystemhowever,thesurfacewasmuchmoreunstablepossiblyduetotherapidoxidationofpurePborthegeneraljaggednessoftheevaporatedPbsurface.ItbettersuitedourworktomovetoaPbBialloyasitgenerallyformedmuchsmoothers82ontheTIsurface.ThismadeitmucheasiertocreatemorestablesurfacesforSTMasasmallamountofairexposurewouldnotthesamplesodramatically.FigureA.2:AFMtopographofPbislandsdepositedonthesurfaceofBi2Se3.Theislandsaregrainy,muchlikethePbBisamples,butalsoareheavilyoxidizedduetotheintrinsicnatureofPbexposedtoair.Assuch,weputagreatdealofintotryingtomeasurethesuperconductingproximityusingthesesamples.ThebestmeasurementfromthisspgeometryisshowninFig.A.3.Thisdataisacleardemonstrationofgap-likedensityofstatesmeasurementsappearingnearPbislands,buttheinducedgapisfartoowidetobeinducedbysuperconductivity.WebelievethattheseDOScurvescouldstemfromchargedensitywavescausebylattice83misalignmentfromthesamplepreparationsteps.Regardless,itwasverytogethighqualitySTMmeasurementsonoraroundthePbislandsduetohighmechanicalinstabilityoftheSTMtipcausedbyunstablePbstructures.FigureA.3:STMdatatakenonasamplemadewithPb/Bi2Se3.Whilewecanseeaprettycleaninterface,theSTSdatashowinducedgapsthatarefartoowidetobesuperconducting.WebelievethatthesegapsstemfromchargedensitywavescausedbylatticemisalignmentoftheTIcrystal.Weexcludedatafromposition4intheimageduetoveryhighmechanicalinstabilityattheinterface.ThisstageofourexperimentmaynothavebeenfruitfulintermsofmeasuringtheinducedsuperconductivityinBi2Se3,butitwascrucialinthatittaughtushowtomakethesamplesconsistentlyaswellaswhatmaterialtousetogetthehighqualitydatashowninChap.4.1.3.84AppendixBTopologicalInsulator/ConventionalInsulator\Dual"ProximityectWehavealreadydemonstratedthatplacingaTIincontactwithanormalmetalorsupercon-ductorcreatesamultitudeoffascinatingphenomenon,butwhatwouldhappenifweputtheTIincontactwithaconventionalinsulator(CI)?AccordingtotheoriginalTItheory,placingaTIincontactwithaCIwouldharbornointerestingphysicssincewestatedinSec.3.1thattheTSSexistsattheboundarybetweenthebulkofaTIandatrivialinsulator(usuallyvacuum).Asitturnsout,thisisnotentirelycorrectanddependsonthespconventionalinsulatorthatweputattheinterface.InthischapterIwilldiscussthetheoreticalpredictionsaswellassomeofourpreliminaryexperimentalresultsonthismatter.TheoryandMotivationWhenplacingaTIincontactwithaCI,whichhasaCherninvariant=0,thetheorypresentinSec.3.1predictsthatthetopologicalsurfacestatewillexistattheinterface.Asitturnsout,moresubtlebehaviorispossible.In2013,Wuet.al.publisheddensityfunctionaltheory(DFT)calculationsdemonstratingthatthetopologicalsurfacestatelocationcanactuallybetuneddependingonwhatmaterialisplacedonthesurface.[2]Inthecaseofthesecalculations,WuusedZnbasedinsulators,eachofwhichharborstpredicted85FigureB.1:FiguretakenfromWuet.al.demonstratingthepossiblebehavioroftheTSSwhenaCIinputincontactwithaTI.TheTIwilleither\topologize"theCI,causingtheTSStotothesurface,becometrivializedbytheCI,pushingtheTSSdownoneQL,orthematerialswillnotoneanotherinanymeaningfulway,leavingtheTSSattheinterface.[2]behaviors.InFig.B.1,wecanseethethreepossiblebehaviors;theTSStothetopoftheCI,theTSSremainsattheinterfacewiththeCI,ortheTSSispusheddownonequintuplelayer(QL)intotheTI.Thisbehaviorisknownasthe\dual"proximityanditdependsnotonlyonthematerialoftheCI,butalsouponthechoiceofTIusedintheexperiment.Inthiscase,thetwopossibleTI'sareeitherBi2Se3orBi2Te3,whiletheCI'scouldbeZnS,ZnSe,orZnTe.TheabilitytotunethelocationoftheTSScouldbeveryusefulinthatwewouldbeabletopossiblypushtheTSSdownonequintuplelayerintotheTI,isolatingitfromsurfacedefectsandallowingforamorepureTSS.Likewise,wecouldalsopulltheTSSfurtherfromthebulkoftheTIandpotentiallyimprovethethermoelectricpropertiesoftheTImaterial.TheDFTcalculationsdonebyWuet.al.byusinga6QLthickTIwithaspCIdepositedonthesurface.AnexampleofthecalculatedresultscanbeseeninFig.B.2.Inthiscase,thetopcurvesshowtheinteractionofZnS/Bi2Se3andthebottomcurvesshowthe86FigureB.2:DFTcalculationtakenfromWuet.al.showingthebandstructuresofTI/CIinterfaces,spZnS/Bi2Se3(a-d)andZnSe/Bi2Se3(e-h).Thedotsindicatethespectralweightsandcontributionsfromatoms,denotedbythesizeandcolorrespectively.DPUandDPLdenotetheDiracpointsoftheupperandlowerssurfacesrespectively.[2]interactionZnSe/Bi2Se3.Asexpectedinbothcases,theTSScontinuestoexistatthebottomoftheTI(the6thQL),butinterestingly,itchangeslocationbasedontheCImaterial.InthecaseofZnSeonBi2Se3,theCIisessentiallymadeintoanotherlayeroftheTI(\topologized")andtheTSSleaksontothesurfaceoftheZnSe.WhereasinthecaseofZnSonBi2Se3,theTSSremainsintheQLattheinterfacebetweenthematerials.ThiscanbeseenclearlyinFig.B.2(d)and(h),wherethechargedensityismappedforeachlayer.Tocompletelysummarizetheresultsofthepaper,wecanlooktoTableB.1.Herewecan87System(Substrate)Vg(eV)a(A)aEb(eV)ZZnS(Bi2Se3)1.82(0.40)6.10(5.55)3.82(4.14)+8.4%0.06InterfaceZnSe(Bi2Se3)1.58(0.40)5.77(5.55)4.00(4.14)+3.5%0.09TopZnTe(Bi2Se3)1.85(0.40)3.98(5.55)4.31(4.14)-3.9%0.29InsideZnTe(Bi2Te3)1.39(0.39)4.93(5.00)4.31(4.38)+1.6%0.19TopTableB.1:TablesummarizingtheresultsoftheWuet.al.paperandlabelingthecharac-teristicpropertiesofthematerialsusedinthecalculations.Vgisthebandgapwidth,istheworkfunction,aisthebulklatticeconstant,aisthelatticemismatch,EbisthebindingenergybetweentheCIandTIlayers,andZisthelocationoftheTSSforthespgeometrycalculated.[2]seethatthemostinterestingresultsoccurwhentheCIisZnSeorZnTeonBi2Se3orZnTeonBi2Te3.Inthisexperiment,weattempttoinvestigatethelocationoftheTSSwhenZnSeisdepositedontothesurfaceofBi2Te3,whichdoesnothaveacalculatedpredictionforthebehavioroftheTSS.ExperimentSamplePreparationOursamples,grownbytheSougroupattheHongKongUniversityofScienceandTechnology,consistof10nmofbulkTIBi2Te3with1-2nmofCIZnSegrownofthesurfaceoftheTI.Thisisallgrownontopofn-dopedGaAsandisshippedtouswitha20nmTecappinglayertoprotectthesurfaceofinterest.BecausetheTecaplayercannotberemovedin-situ,wehavedevelopedasamplepreparationmethodinwhichtheTecaplayerisremovedandthesampleremainsinnitrogengasorvacuumforthedurationoftheexperiment.Inordertohandlethesampleeasier,weattachittoalargerpieceofundopedGaAsviaInsolder.Theisdonebyheavilyscratchingthebottomofthen-dopedGaAschipinordertoremovetheoxidelayerandallowtheIntoestablishgoodelectricalcontactwiththe88sample.WethenplacethelargeundopedGaAschiponahotplatewithalargeamountofInsolderonthesurface,wherewethensandwichtheInbetweenthesampleandtheGaAschip.Afterthisiscooled,wethenloadthesampleintoanovenandbakeitat315Cfor20minutesinanitrogengasenvironment.InordertodemonstratethattheTecaplayerwassuccessfullyremoved,wedidAFMandSTMonabakedandunbakedsample,asshowninFig.B.3.HerewecanseetwoclearlyregionswhichwebelievetobetheZnSeterracesandthebareBi2Te3surface.Wealsodidenergy-dispersiveX-rayspectroscopy(EDS)toclassifythematerialsexposedonthesurfacebeforeandafterbaking,whichdemonstratedthattheTecaplayerwascompletelyremovedbythebakingprocess.Afterbaking,thesampleisthenremovedfromtheovenandsilverpaintedonthemetalsamplediscformeasurement.AsilverpaintbridgeisalsomadefromtheInsoldertothesamplediscinordertoensuregoodelectricalcontact.Inthiscase,theresistancebetweenthesamplesurfaceandthesamplediscisapproximately5katroomtemperature,whichisindicativeofgoodconductivityforthepurposesofSTM.AdiagramshowingtheresultcanbeseeninFig.B.4.Afterthesampleisprepared,itisloadedintothemicroscopeandcooledto77K.MeasurementSystemThesystemweusetoconductSTMandSCAimagingonoursamplesisthehousebuiltsystemshownanddiscussedinChapter2.ByutilizingourHEMTcircuit,weareabletofreelychangebetweenSTMandSCAwithease,allowingustogetmanymeasurementsandmaximizeourabilitytocharacterizethephysicsweseeduringthismeasurement.ThegeometryoftheexperimentisdemonstratedinFig.B.5.WehavevariousterracesonZnSeonthesurface,whichwewillsituatethescanningprobeoverandsweepthebiasvoltage.WhentheTSSchangeslocation,wewillseeachangeincapacitance,whichwill89FigureB.3:(a)AFMphaseretracedemonstratingtmaterialsonthesurfaceofabakedsample.(b)STMtopographofthesample,demonstratingthesamefeaturesseenintheAFMscanshownin(a).FigureB.4:Diagramshowingsampleafterbakingandassembly.Thesampleitselfisusuallyaround3mminsizeinordertoallowforalargeareatoscanusingoursurfaceprobes.90translatetoastepinthecompressibilitymeasurement.FigureB.5:(a)TheelectricbetweenthetipandtheTSSintheZnSeterracehasasetcapacitanceCZnSe.(b)TheelectricbetweenthetipandtheTSSintheBi2Te3layerwithacapacitanceofCBi2Te3.Herewewouldseeasmallercapacitancebecauseofthelargerdistancebetweenthetwocapacitive\plates".ToreiteratethemostimportantequationforSCAmeasurements,SCAissensitivetotwoveryimportantphysicalphenomenawhenmeasuringcompressibility;thegeometriccapacitance(Cgeo)andthethermodynamicdensityofstates(dN).Thesearerelatedtothemeasuredcapacitance(Cm)bytheequationACm=ACgeo+1e2dN(B.1)91Whenthegeometriccapacitanceremainsconstant,Cmbecomessensitivetothethermody-namicdensityofstates.[4]ResultsSofarwehavebeenabletogetmanyresultsusingSCAat77KbothonandZnSeterraces.WedothisbylocatinganidealspotonthesampleusingSTM,thenswitchingtoSCAtomeasurethecompressibilityatthedesignatedlocation.FormeasurementstakenoverbareBi2Te3,whichareshowninFig.B.6,weseewhatwebelievetobetheDiracconeintherangeof1{1.5V.ThisisindicativeofthetopologicalsurfacestatestronglyexistinginthesurfaceoftheTIlayer,asisexpected.Thepotentialatthesurfacestateislessthanthepotentialappliedbythetip.Thisisoftenreferredtoastheleverarm,whichweestimatetobeapproximatelyafactorof10.ItshouldalsobenotedthatincompressibilitymeasurementswewillnotseeafullDiracconeduetothewaytheDOScontributestothecapacitance.SCAissensitivetotheDOSonlywhentheDOSterminEq.B.1becomessmallcomparedtothegeometriccapacitanceterm.ThismeansthatwemayonlybeabletoseeadipinthecompressibilitywheretheDiracconeDOSisatitssmallest.InFig.B.7weshowcompressibilitymeasurementstakenonvariousZnSeterraces.WhiletherearehintsofDiracconebehaviorandcapacitancesteps,wehavenotbeenabletoresolvesuchfeaturesreproducibly.Thesignalofthesefeaturesistoosmallcomparedtothenoise.TheseresultsindicatethattheTSSiseitherremainingattheinterfacebetweentheCIandtheTIorthattheTSSismovingdeeperintothesampleandthesignalisbeingscreenedbyotherelectronicstateshigherupinthesamplesurface.Wehavetakentensofthousandsofmeasurementsontheselocations,butthusfarhavebeenunabletoseesignsofaTSSorastepfromthetopsurfaceoftheCIdowntotheCI/TIinterface.92FigureB.6:CompressibilitymeasurementtakenovervariousbareBi2Te3at77K.Whilethereisanotableslopepresentinthesemeasurements,weseewhatwebelievetobetheDiracconenear1{1.5Vinthesecurves.ThiswouldthekeysignatureofthetopologicalsurfacestateandisexpectedwhenoverabareTI.Allcurvesshownareaveragesofthousandsofindividualcurvesinordertosuppressnoiseandenhancethesignal.Theblackcurvesaretherawdatawhiletheredcurvesaretheresultsofsmoothingtherawdata.93FigureB.7:CompressibilitymeasurementtakenovervariousZnSeterracesat77K.Asidefromtheapparentslope,therearenonotablefeaturesinthesemeasurements.ThelackofastepinthecapacitanceorappearanceoftheDiracconeindicatesthatthereisnoinducedtopologicalsurfacestateinthisspinterface.Allcurvesshownareaveragesofthousandsofindividualcurvesinordertosuppressnoiseandenhancethesignal.94Ofcourse,wealsoutilizedourabilitytodoSTMandSTSwhileusingourSCAIcircuit.At77K,STSisverytododuetothepoorconductivityoftheGaAssubstrate.Howeveratroomtemperature,wewereabletoacquiresomespectroscopyonthebareBi2Te3.FigureB.8:(a)STMtopographshowingthecleartregionsofZnSeandBi2Te3.(b)STMspectroscopymeasurementtakenonbareBi2Te3showingtheusualdensityofstatesforaDiracconeatroomtemperature.Inanevenfurtherattempttocharacterizewhatweseeinthesesamples,wealsoconductedmultipleKelvinprobemeasurementsinordertoseeifthesurfacepotentialofthematerialhadanyonthecompressibility.PleaserefertoAppendixCforadiscussiononthemethodologyusedinourKelvinprobemeasurements.Fig.B.9showsbothaKelvinprobetopograph(a)andthecorrespondingnullingvoltagemeasurementstakeninthebrightanddarkregions(b,c).TheresultingVNinthedarkregionwas-1.035Vandinthelightregionwas-1.132V.ItwouldhavebeenhighlyinstructiveastothenatureofthesemeasurementstotakeSTMspectraintheseregions.However,at77K,itwasimpossibletotakehighqualitytunnelingspectroscopyduetopoorthepoorconductanceofthesamplesatlowtemperatures.95FigureB.9:(a)Kelvinprobetopographshowingregionswithsurfacepotentials.PleaserefertoAppendixCforadiscussiononhowthenullingvoltageVNisacquired.(b)Kelvinprobespectrumtakeninthedarkregionatthebottomof(a).TheresultingnullingvoltageVNis-1.035V.(c)Kelvinprobespectrumtakeninthebrightregionof(a).Here,VNis-1.132V.WhileitwouldhavebeeninterestingtotakeSTMspectroscopyintheseregionstoseeiftheDiracconeshifted,itwasnotpossiblewiththissampleduetopoorconductivityat77K.96ConclusionsWehavesuccessfullyfoundawaytopreparethesamplesandloadthemintothemicroscope.ThereisnodoubtthatourpreparationproceduresdoexactlywhatweanticipatedduetothefactthatweseetheexpectedstructureonthesurfaceandareabletomeasuretheDiracconeusingSTM.Whenmeasuringcompressibility,unfortunatelywedonotresolveanynoticeablefeaturesorseeanystepsincapacitance,indicatingthatthecombinationofZnSeandBi2Te3resultsinastationaryTSSattheinterfacebetweenthesetwomaterialsoratleastastepdownintothesamplewhichwecannotresolveduetoelectronicscreeningHowever,webelievethatweareseeingtheDiracconestateswhenmeasuringthecompressibilityoverbareBi2Te3,whichimpliesthatweshouldalsoseetheDiracconeontheZnSeterracesifitwereindeedinthisgeometry.DuetoOhmiccontactissues,weareunabletoconductSTSscansatlowtemperature,butareabletoatleastdospectroscopyatroomtemperature,meaningthatthermalexcitationsareneededtoreachoptimalvoltagesatthesamplesurface.Wehavesetthegroundworkforfurtherstudiesofthedualproximityofconventionalinsulator/topologicalinsulatorinterfaces.Whiletheworkpresentedhereishighlypreliminaryintermsofmeasuringatunablelocationforthetopologicalsurfacestate,webelievethatothersamplematerialcombinationscouldleadtothepredictedresultsoftheTSSchanginglocation.Weobservedrepeatablecone-likefeaturesoverbareBi2Te3whichwebelievetobethesignaturesoftheDiraccone,whereasontheZnSeterraces,wesawnosignofthisconefeatureaswellasnostepinthecapacitance.Thisleadsustoconcludethatthetopologicalsurfacestateremainsattheinterfacebetweenthesematerials.Ifthetheoreticalpredictionsarelaterrevealedtobecorrect,theimplicationsofatunablesurfacestatelocationcouldleadtohighlydevicesinwhichtheTSSwillbeprotectedfromdisorder.97AppendixCKelvinProbeStemmingfromourstrongcapabilitiestomeasurechargeaccumulationatthetipapex,ourscanningprobesystemscanalsoperformameasurementknownasKelvinprobe.KelvinprobeisrelatedtoourSCAImeasurementsthroughtheuseofthesameHEMTcircuitmentionedinsection2.2,wheretheprimarybetweenthetwomeasurementsisthatthetiposcillatesabovethesampleinsteadofthebiasvoltageoscillating;inthiscaseweuseapurelyDCbias.Here,thecircuitisonceagainsensitivetotheelectricbetweenthetipandthesample,butthemeasurementisnolongerinvestigatingsubsurfacecharges.Kelvinprobeisusedprimarilytomeasurethesurfacepotential,orworkfunction,ofthesample.Foreaseofdiscussion,wewillcallthispotentialthenullingvoltage(VNull)inthefollowingparagraphs.InKelvinprobe,themeasuredchargeisdirectlysensitivetothedistancebetweenthetipandthesample.Thismeasurementutilizestwoprimarydistances,themeandistance,z0,whichisthedistanceofthetipfromthesamplesurfacewithoutanyoscillatorybehavior,andtheoscillationdistance,z,whichistheamplitudeoftheoscillatorydistancebetweenthetipandthesample.Thelock-inisusedtomeasurethechargeaccumulatedonthetip,C,asafunctionofDCbiasvoltageV.Whenthetipisclosertothesample,thelock-inwilldetectalargersignalduetoahigherconcentrationofelectricdterminationsonthetipasopposedtowhenthetipisfurtheraway.Thuswecanutilizethe98wellknownequationQ=CVtoexplainthephysicsofthismeasurement.Inthespcaseforthismeasurement,wehaveQ=C(VVNull)+Q0(C.1)whereCisthecapacitancebetweenthetipandthesampleandQ0issomeconstant"charge"stemmingfromthescanningprobeelectronics,whichfunctionsassomearbitrarysetofnoimportancetothemeasurement.Wecanfurtherdescribethecapacitancebyutilizingaparallelplatecapacitorapproximation,whichgivesustheequationQ=0z0+z(VVNull)+Q0(C.2)whereAistheeareaofthetipand0isthepermittivityoffreespace.Thelock-inisonlysensitivetotheoscillatorybehaviorofthemeasurement,soequationC.2downtoQLockIn=0zz20(VVNull)+Q0(C.3)Fromhere,wecanclearlyseethatVNullcanbemeasuredbythepointatwhichQLockIn=Q0.However,thisisimpossibletodousingonlyonecurve.Themeasurementisdirectlysensitivetothemeandistancez0,butregardlessofwhatdistanceexistsbetweenthetipandthesample,thepointatwhichQLockIn=Q0willoccuratthesamevoltageV.Thismeansthatbytakingthreemeasurementsatvaluesofz0,wecanmeasureVNullwithoutactuallyneedingtoknowthevalueofQ0.WherethethreeseparatemeasurementsintersectonaplotofQLockInvs.V,wewillhavethevalueforVNull.AplotdemonstratingthiscanbeseeninFig.C.1.99FigureC.1:AplotsimulatingthreedC=dzmeasurementstakenatthesamepointattvaluesofz0.Herewecanseethecrossingpointoccursatavoltageof.5Vandatameasuredvalueof4fordC=dz.Thistellsusthatthenullingvoltagewouldthenbe.5Vandthearbitraryfromthemeasurementelectronicshasavalueof4.100Ofcourse,duringtheactualexperiment,theHEMTandtiphaveaseparateappliedvoltage,Vtip,whichiscrucialfortuningtheHEMTduringthemeasurement.ThisadjustsourresultsbytherelationV=VNullVtip.InordertogettheVNull,wesimplyneedtoaddVtiptoVextractedfromthedC=dzvs.Vplotpreviouslymentioned.ThevalueofVTipistypicallyaround-.4V,butitisimportanttorecordtheexactvaluewhentakingthemeasurementsinorderhavethebestaccuracypossible.101BIBLIOGRAPHY102BIBLIOGRAPHY[1]J.Bardeen,L.N.Cooper,andJ.R.ScPhysRev.108,1175(1957).[2]G.Wu,H.Chen,Y.Sun,X.Li,P.Cui,C.Franchini,J.Wang,X.-Q.Chen,andZ.Zhang,Sci.Rep.3,1233(2013).[3]K.Besocke,SurfaceScience,181,145(1987).[4]S.H.Tessmer,P.I.Glicofridis,R.C.Ashoori,L.S.Levitov,andM.R.Melloch,Nature395,6703(1998).[5]K.Walsh,M.Romanowich,M.Gasseller,I.Kuljanishvili,R.Ashoori,andS.H.Tessmer,JVisExp.77,50676(2013).[6]M.Z.HasanandC.L.Kane,Rev.Mod.Phys.82,3045(2010).[7]D.Thouless,M.Kohmoto,M.Nightingale,andM.Dennijs,PhysicalReviewLetters49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